Conditionally positive definiteness in operator theory
aa r X i v : . [ m a t h . F A ] A ug Conditionally positive definiteness in operator theory
Zenon Jan Jab lo´nski, Il Bong Jung, and Jan Stochel
Abstract.
In this paper we extensively investigate the class of conditionallypositive definite operators, namely operators generating conditionally positivedefinite sequences. This class itself contains subnormal operators, 2- and 3-isometries and much more beyond them. Quite a large part of the paper isdevoted to the study of conditionally positive definite sequences of exponen-tial growth with emphasis put on finding criteria for their positive definiteness,where both notions are understood in the semigroup sense. As a consequence,we obtain semispectral and dilation type representations for conditionally pos-itive definite operators. We also show that the class of conditionally positivedefinite operators is closed under the operation of taking powers. On the basisof Agler’s hereditary functional calculus, we build an L ∞ ( M )-functional calcu-lus for operators of this class, where M is an associated semispectral measure.We provide a variety of applications of this calculus to inequalities involv-ing polynomials and analytic functions. In addition, we derive new necessaryand sufficient conditions for a conditionally positive definite operator to be asubnormal contraction (including a telescopic one). Contents
1. Preliminaries 21.1. Introduction 21.2. Notation and terminology 52. Conditionally positive definite sequences 62.1. Basic facts 62.2. Exponential growth 72.3. Additional constraints 153. Representations of conditionally positive definite operators 193.1. Semispectral integral representations 193.2. A dilation representation 243.3. A simplified representation with applications 293.4. Subnormality 364. A functional calculus and related matters 41
Mathematics Subject Classification.
Primary 47B20, 44A60; Secondary 47A20, 47A60.
Key words and phrases.
Conditional positive definiteness, positive definiteness, subnormality,functional calculus.The research of the second author was supported by Basic Science Research Programthrough the National Research Foundation of Korea (NRF) funded by the Ministry of Educa-tion (2018R1A6A3A01012892).
1. Preliminaries1.1. Introduction.
The concepts of positive and conditional positive defi-niteness (at least in the group setting) have their origins in stochastic processesthat are stationary or which have stationary increments [
41, 46, 49, 14, 55 ]. Itseems that conditional positive definiteness appeared in operator theory for thefirst time on the occasion of investigating subnormal operators (see [ ]). Later itappeared sporadically in the context of complete hyperexpansivity and completehypercontractivity of finite order, both related to m -tuples of commuting operators[
9, 20, 21 ]. The main goal of the present paper is to exploit conditional posi-tive definiteness in the semigroup setting to study a class of operators which islarge enough to subsume subnormal operators (which are integrally tied to positivedefiniteness), 2- and 3-isometries, certain algebraic operators which are neither sub-normal nor m -isometric, and much more. Below we give a more detailed discussionon this.Throughout this paper H stands for a (complex) Hilbert space and B ( H ) forthe C ∗ -algebra of all bounded linear operators on H . Recall that an operator T ∈ B ( H ) is said to be subnormal if there exist a Hilbert space K and a normaloperator N ∈ B ( K ), called a normal extension of T , such that H ⊆ K (isometricembedding) and
T h = N h for all h ∈ H . The celebrated Lambert’s characterizationof subnormality [ ] can be adapted to the context of not necessarily injectiveoperators as follows (for (i) ⇔ (ii) see [ , Theorem 7], while for (ii) ⇔ (iii) applyTheorem 2.1.3 substituting T h in place of h ). Theorem . If T ∈ B ( H ) , then the following conditions are equivalent :(i) T is subnormal, (ii) the sequence {k T n h k } ∞ n =0 is a Stieltjes moment sequence for every h ∈ H , (iii) the sequence {k T n h k } ∞ n =0 is positive definite for every h ∈ H . The above theorem, which fails for unbounded operators (see [
37, 17 ]), turnedout to be very useful when studying the concrete classes of bounded operators (see[
45, 36, 16, 66, 18 ]). Some of them are associated with the set F of noncon-stant entire functions with nonnegative Taylor’s coefficients at 0. The questionof characterizing subnormality of composition operators with matrix symbols on L ( R d , ρ ( x ) dx ) with a density function ρ coming from Φ ∈ F (see [ ]) led tothe following problem, which for almost thirty years remains unsolved even forsecond-degree monomials (see [ , p. 237]). Problem . Let T ∈ B ( H ) be a contraction and Φ ∈ F . Is it true that if { Φ ( k T n h k ) } ∞ n =0 is a Stieltjes moment sequence for all h ∈ H , then T is subnormal ?If T is contractive, the answer to the question in Problem 1.1.2 is in the af-firmative as long as Φ ′ (0), the derivative of Φ at 0, is positive or T is algebraic(see [ , Theorems 5.1 and 6.3]). If T is not contractive, then the question has a ONDITIONALLY POSITIVE DEFINITENESS IN OPERATOR THEORY 3 negative answer (see [ , Example 5.4]). Note also that the converse implication inProlem 1.1.2 is true even if T is not contractive (see the proof of [ , Theorem 5.1]).If T ∈ B ( H ) and Φ = exp (which is a member of F with Φ ′ (0) > { exp( k T n h k ) } ∞ n =0 is a Stieltjes moment sequence for every h ∈ H if and only if the sequence {k T n h k } ∞ n =0 is conditionally positive definite for every h ∈ H (see Subsection 2.1 for definition). The situation becomes more complex ifthe function exp is replaced by an arbitrary member Φ of F ; then the hypothesisthat { Φ ( k T n h k ) } ∞ n =0 is a Stieltjes moment sequence for every h ∈ H implies thatfor some positive integer j (depending only on Φ ) and for every h ∈ H , the sequence {k T n h k j } ∞ n =0 is conditionally positive definite (see [ , Lemma 5.2]). It was shownin [ , Theorem 4.1] that if T is a contraction, then T is subnormal if and only ifthe sequence {k T n h k } ∞ n =0 is conditionally positive definite for every h ∈ H . Thecontractivity hypothesis cannot be removed (see [ , Example 5.4]).The above-mentioned results of [ ] were obtained by using ad hoc methods.The main goal of the present paper is to systematically and rigorously study opera-tors T ∈ B ( H ) having the property that for every h ∈ H , the sequence {k T n h k } ∞ n =0 is conditionally positive definite. Such operators are called here conditionally pos-itive definite. In view of Theorem 1.1.1 and the fact that positive definite se-quences are conditionally positive definite, subnormal operators are conditionallypositive definite but not conversely. Among prominent examples of non-subnormalconditionally positive definite operators are non-isometric 3-isometries (see Proposi-tion 4.3.1 and [ , Proposition 4.5]). Our investigations are preceded by developingharmonic analysis of conditionally positive definite functions of (at most) exponen-tial growth on the additive semigroup of nonnegative integers. As a consequence,we gain, among other things, a deeper insight into the subtle relationship betweensubnormality and conditional positive definiteness.The organization of this paper is as follows. We begin by introducing notationand terminology in Subsection 1.2 and collecting more or less known facts aboutpositive and conditionally positive definite (scalar) sequences in Subsection 2.1.The remainder of Section 2 is devoted to systematic study of conditionally positivedefinite sequences. In Subsection 2.2 we provide an integral representation for aconditionally positive definite sequence of exponential growth and relate the rateof its growth to the “size” of the closed support of its representing measure (seeTheorem 2.2.5). We also compare the integral representations for positive definiteand conditionally positive definite sequences (see Theorem 2.2.12). Theorem 2.2.13,which is the main result of this subsection, states that a sequence { γ n } ∞ n =0 ofexponential growth is positive definite if and only if 0 is an accumulation pointof the set of all θ ∈ (0 , ∞ ) for which the sequence { θ n γ n } ∞ n =0 is conditionallypositive definite. In Subsection 2.3 we characterize conditionally positive definitesequences of exponential growth for which the sequence of consecutive differencesis either convergent or bounded from above plus some additional constraints (seeTheorems 2.3.2 and 2.3.3). As a consequence, we show that, subject to somemild constraints, convergent conditionally positive definite sequences of exponentialgrowth are positive definite (see Corollary 2.3.4).Starting from Section 3, we begin the study of conditionally positive definiteoperators. In Subsection 3.1 we give a semispectral integral representation for such Z. J. JAB LO´NSKI, I. B. JUNG, AND J. STOCHEL operators and relate their spectral radii to the closed supports of representing semis-pectral measures (see Theorem 3.1.1). Certain semispectral integral representationsfor completely hypercontractive and completely hyperexpansive operators of finiteorder appeared in [
35, 20, 21 ] with the representing semispectral measures concen-trated on the closed interval [0 , Q (see Proposition 3.3.5).In both cases, we describe explicitly the corresponding semispectral integral anddilation representations. In Theorem 3.4.1 we give necessary and sufficient condi-tions for a conditionally positive definite operator T to be subnormal written interms of the semispectral integral representation of T . Theorem 3.4.4, which isthe main result of Subsection 3.4, provides several characterizations of subnormalcontractions via conditional positive definiteness including the one appealing to thetelescopic condition. This is a generalization of [ , Theorem 4.1]. On the basis ofearlier results, we characterize conditional positive definiteness of a (bounded) op-erator T on H by subnormality of (in general unbounded) unilateral weighted shifts W T,h , h ∈ H , canonically associated with T (see Proposition 3.4.9). If the sequence { T ∗ ( n +1) T n +1 − T ∗ n T n } ∞ n =0 is not convergent in the weak operator topology, thensome (or even all except h = 0) weighted shifts W T,h may be unbounded. If thelimit exists and is nonzero, then all W T,h are bounded, but T is not subnormal.Finally, if the limit exists and is equal to zero, then T is a subnormal contraction.In Subsection 4.1, we construct an L ∞ ( M )-functional calculus for conditionallypositive definite operators (see Theorem 4.1.2). As a consequence, we obtain avariety of estimates on norms of polynomial and analytic expressions coming fromoperators in question (see Corollary 4.1.3 and Subsection 4.2). The last subsectionof this paper is devoted to characterizing conditionally positive definite operatorsfor which the closed support of the associated semispectral measure is one of thethree sets ∅ , { } and { } . It is shown that the first two cases completely characterizeconditionally positive definite m -isometries (see Proposition 4.3.1). The third caseleads to conditionally positive definite operators that are beyond the classes ofsubnormal and m -isometric operators (see Proposition 4.3.5 and Example 4.3.6).The research on conditionally positive definite operators initiated in this paperhas a continuation. Namely, we have found the description of algebraic condition-ally positive definite operators, characterizations of conditionally positive definiteweighted shifts, which are a good source of examples and counterexamples, as wellas some answers to the question of similarity to subnormal operators. We intendto include these topics in separate papers. ONDITIONALLY POSITIVE DEFINITENESS IN OPERATOR THEORY 5
We denote by R and C the fields of realand complex numbers, respectively. Since we consider suprema of subsets of R which may be empty, we adhere to the often-used convention thatsup ∅ = sup x ∈∅ f ( x ) := −∞ whenever f : R → R . (1.2.1)We write N , Z + and R + for the sets of positive integers, nonnegative integersand nonnegative real numbers, respectively. As usual, C [ X ] stands for the ringof all polynomials in indeterminate X with complex coefficients. We customarilyidentify members of C [ X ] with polynomial functions of one real variable. Theunique involution on C [ X ] which sends X to itself is denoted by ∗ , that is, if p = P i > α i X i ∈ C [ X ], then p ∗ = P i > α i X i , or in the language of polynomialfunctions p ∗ ( x ) = p ( x ) for all x ∈ R . If no ambiguity arises, the characteristicfunction of a subset Ω of a set Ω is denoted by χ Ω . Given a compact topologicalHausdorff space Ω, let C ( Ω ) stand for the Banach space of all continuous complexfunctions on Ω with the supremum norm k f k C ( Ω ) = sup x ∈ Ω | f ( x ) | , f ∈ C ( Ω ) . We write B ( Ω ) for the σ -algebra of all Borel subsets of a topological Hausdorffspace Ω. If not stated otherwise, measures considered in this paper are assumed tobe positive. The closed support of a finite Borel measure µ on R (or C ) is denotedby supp( µ ) (recall that supp( µ ) exists because µ is automatically regular, see [ ,Theorem 2.18]). Given x ∈ R , we write δ x for the Borel probability measure on R such that supp( δ x ) = { x } .Given (complex) Hilbert spaces H and K , we denote by B ( H , K ) the Banachspace of all bounded linear operators from H to K . We abbreviate B ( H , H ) to B ( H ) and denote by B ( H ) + the convex cone { T ∈ B ( H ) : T > } of nonnegativeoperators on H . We write I H (or simply I if no ambiguity arises) for the identityoperator on H . Let T ∈ B ( H ). In what follows, N ( T ), R ( T ), σ ( T ), σ p ( T ), r ( T ) and | T | stand for the kernel, the range, the spectrum, the point spectrum,the spectral radius and the modulus of T , respectively. To comply with Gelfand’sformula for spectral radius, we adhere to the convention that r ( T ) = 0 if H = { } . We say that T is normaloid if r ( T ) = k T k , or equivalently, by Gelfand’s formulafor spectral radius, if and only if k T n k = k T k n for all n ∈ N . Let us recall thefollowing basic fact (see [ , Proposition II.4.6], see also [ , p. 116]). Any subnormal operator is normaloid. (1.2.2)This will be used several times in this article. Given an operator T ∈ B ( H ), we set B m ( T ) = m X k =0 ( − k (cid:18) mk (cid:19) T ∗ k T k , m ∈ Z + . (1.2.3)If m ∈ N and B m ( T ) = 0, then T is called an m -isometry (see [ , p. 11] and[
3, 4, 5 ]). An m -isometry T is said to be strict if m = 1 and H 6 = { } , or m > T is not an ( m − H 6 = { } (see [ ]). Examples ofstrict m -isometries for each m > , Proposition 8]. We say that T is 2 -hyperexpansive if B ( T ) ]). We call T completely hyperexpansive if B m ( T ) m ∈ N (see [ ]).Let F : A → B ( H ) be a semispectral measure on a σ -algebra A of subsetsof a set Ω , i.e., F is σ -additive in the weak operator topology (briefly, wot ) and Z. J. JAB LO´NSKI, I. B. JUNG, AND J. STOCHEL F ( ∆ ) > ∆ ∈ A . Denote by L ( F ) the linear space of all complex A -measurable functions ζ on Ω such that R Ω | ζ ( x ) |h F (d x ) h, h i < ∞ for all h ∈ H .Then for every ζ ∈ L ( F ), there exists a unique operator R Ω ζ d F ∈ B ( H ) such that(see e.g., [ , Appendix]) D Z Ω ζ d F h, h E = Z Ω ζ ( x ) h F (d x ) h, h i , h ∈ H . (1.2.4)If Ω = R , C and F : B ( Ω ) → B ( H ) is a semispectral measure, then its closed sup-port is denoted by supp( F ) (recall that such F is automatically regular so supp( F )exists). By a semispectral measure of a subnormal operator T ∈ B ( H ) we mean anormalized compactly supported semispectral measure G : B ( C ) → B ( H ) definedby G ( ∆ ) = P E ( ∆ ) | H for ∆ ∈ B ( C ), where E : B ( C ) → B ( K ) is the spectral mea-sure of a minimal normal extension N ∈ B ( K ) of T and P ∈ B ( K ) is the orthogonalprojection of K onto H (the minimality means that K has no proper closed vec-tor subspace that reduces N and contains H ). It follows from [ , Proposition 5]and [ , Proposition II.2.5] that a subnormal operator has exactly one semispectralmeasure. It is also easily seen that T ∗ n T n = R C | z | n G (d z ) for all n ∈ Z + . Applying(1.2.4) and the measure transport theorem (cf. [ , Theorem 1.6.12]) yields T ∗ n T n = Z R + x n G ◦ φ − (d x ) , n ∈ Z + , (1.2.5)where φ : C → R + is defined by φ ( z ) = | z | for z ∈ C and G ◦ φ − : B ( R + ) → B ( H )is the semispectral measure defined by G ◦ φ − ( ∆ ) = G ( φ − ( ∆ )) for ∆ ∈ B ( R + ).We refer the reader to [ ] for the foundations of the theory of subnormal operators.
2. Conditionally positive definite sequences2.1. Basic facts.
Let γ = { γ n } ∞ n =0 be a sequence of real numbers. Thesequence γ is said to be positive definite if for all finite sequences λ , . . . , λ k ∈ C , k X i,j =0 γ i + j λ i ¯ λ j > . (2.1.1)If (2.1.1) holds for all finite sequences λ , . . . , λ k ∈ C such that P kj =0 λ j = 0,then we say that γ is conditionally positive definite . It is a matter of routine toverify that γ is positive definite (resp., conditionally positive definite) if and onlyif (2.1.1) holds for all finite sequences λ , . . . , λ k ∈ R (resp., for all finite sequences λ , . . . , λ k ∈ R such that P kj =0 λ j = 0). It follows from definition that if γ is positivedefinite (resp., conditionally positive definite), then so is the sequence { γ n +2 k } ∞ n =0 for every k ∈ Z + . However, it may happen that γ is positive definite but { γ n +1 } ∞ n =0 is not (e.g., γ n = ( − n for n ∈ Z + ).The following fundamental characterization of conditional positive definitenessin terms of positive definiteness is essentially due to Schoenberg. Lemma , Lemma 1.7], [ , Theorem 3.2.2]) . If γ = { γ n } ∞ n =0 is asequence of real numbers, then the following conditions are equivalent :(i) γ is conditionally positive definite, (ii) { e tγ n } ∞ n =0 is positive definite for every positive real number t . ONDITIONALLY POSITIVE DEFINITENESS IN OPERATOR THEORY 7
A sequence γ = { γ n } ∞ n =0 of real numbers is said to be a Hamburger (resp.,
Stieltjes , Hausdorff ) moment sequence if there exists a Borel measure µ on R (resp., R + , [0 , γ n = Z t n dµ ( t ) , n ∈ Z + . (2.1.2)A Borel measure µ on R satisfying (2.1.2) is called a representing measure of γ .If γ is a Hamburger moment sequence which has a unique representing measureon R , then we say that γ is determinate . Note that by [ , Ex. 4(e), p. 71], theWeierstrass theorem (see [ , Theorem 7.26]) and the Riesz representation theorem(see [ , Theorem 2.14]) the following holds. Lemma . A Hamburger moment sequence γ = { γ n } ∞ n =0 of real numbers hasa compactly supported representing measure if and only if θ := lim sup n →∞ | γ n | /n < ∞ . Moreover, if this is the case, then γ is determinate and supp( µ ) ⊆ [ − θ, θ ] , where µ is a unique representing measure of γ . In particular, a Hausdorff moment sequence is always determinate. For ourlater needs, we recall a theorem due to Stieltjes.
Theorem ],[ , Theorem 6.2.5]) . A sequence { γ n } ∞ n =0 ⊆ R is aStieltjes moment sequence if and only if the sequences { γ n } ∞ n =0 and { γ n +1 } ∞ n =0 are positive definite. We refer the reader to [
10, 59 ] for the fundamentals of the theory of momentproblems.
In this subsection we give an integral represen-tation for conditionally positive definite sequences of (at most) exponential growth(see Theorem 2.2.5). Positive definite sequences of exponential growth are char-acterized by means of parameters appearing in the above-mentioned integral rep-resentation (see Theorem 2.2.12). Theorem 2.2.13 states that a sequence { γ n } ∞ n =0 of exponential growth is positive definite if and only if the sequences { θ n γ n } ∞ n =0 , θ ∈ R , are conditionally positive definite.We begin by introducing the difference transformation △ which plays an im-portant role in further considerations. Denote by C Z + the complex vector space ofall complex sequences { γ n } ∞ n =0 with linear operations defined coordinatewise. Thedifference transformation △ : C Z + → C Z + is given by( △ γ ) n = γ n +1 − γ n , n ∈ Z + , γ = { γ n } ∞ n =0 ∈ C Z + . Clearly, △ is a linear. Denote by △ k the k th composition power of △ , i.e., △ isthe identity transformation of C Z + and △ k +1 γ = △ k ( △ γ ) for γ = { γ n } ∞ n =0 ∈ C Z + .Given n ∈ Z + , we define the polynomial Q n ∈ C [ X ] by Q n ( x ) = ( x ∈ R and n = 0 , , P n − j =0 ( n − j − x j if x ∈ R and n = 2 , , , . . . . (2.2.1)Below, for a fixed x ∈ R , we write △ Q ( · ) ( x ) to denote the action of the transfor-mation △ on the sequence { Q n ( x ) } ∞ n =0 . Lemma . The polynomials Q n have the following properties : Q n ( x ) = x n − − n ( x − x − , n ∈ Z + , x ∈ R \ { } , (2.2.2) Z. J. JAB LO´NSKI, I. B. JUNG, AND J. STOCHEL Q n +1 ( x ) = xQ n ( x ) + n, n ∈ Z + , x ∈ R , (2.2.3) Q n ( x ) n Q n +1 ( x ) n + 1 , n ∈ N , x ∈ [0 , , (2.2.4)lim n →∞ Q n ( x ) n = 11 − x , x ∈ [0 , , (2.2.5)( △ Q ( · ) ( x )) n = ( if n = 0 , x ∈ R , P n − j =0 x j if n ∈ N , x ∈ R , (2.2.6)( △ Q ( · ) ( x )) n = x n , n ∈ Z + , x ∈ R . (2.2.7) Proof.
Suppose n >
2. Then x n − − n ( x − x − = ( P n − i =0 x i ) − nx − n − X i =0 x i − x − n − X i =1 i − X j =0 x j = n − X j =0 ( n − j − x j , x ∈ R \ { } . This implies (2.2.2). The identities (2.2.3), (2.2.6) and (2.2.7) follow from (2.2.1)and the definition of △ , while (2.2.4) and (2.2.5) can be deduced from (2.2.2). (cid:3) Below, we denote by | µ | the total variation measure of a complex Borel measure µ on R . Recall that a complex Borel measure on R is automatically regular, i.e.,its total variation measure is regular (see [ , Theorem 2.18]). Lemma . Suppose a, b, c ∈ C and µ is a complex Borel measure on R suchthat µ ( { } ) = 0 , the measure | µ | is compactly supported and a + bn + cn + Z R Q n ( x )d µ ( x ) = 0 , n ∈ Z + . Then a = b = c = 0 and µ = 0 . Proof.
Define γ ∈ C Z + by γ n = a + bn + cn + R R Q n ( x )d µ ( x ) for n ∈ Z + . Itfollows from (2.2.7) that0 = ( △ γ ) n = Z K x n d( µ + 2 cδ )( x ) , n ∈ Z + , where K := supp( | µ + 2 cδ | ) is a compact subset of R . This implies that Z K p ( x )d( µ + 2 cδ )( x ) = 0 , p ∈ C [ X ] . Applying the Weierstrass theorem and the uniqueness part in the Riesz Represen-tation Theorem (see [ , Theorem 6.19]), we deduce that ( µ + 2 cδ )( ∆ ) = 0 for all ∆ ∈ B ( R ). Substituting ∆ = { } , we get c = 0, and consequently µ = 0. Clearly, a = γ = 0. Putting all this together gives b = 0, completing the proof. (cid:3) Now, for the reader’s convenience we state explicitly the fundamental charac-terization of conditionally positive definite sequences. Recall that a Borel measureon a Hausdorff topological space is said to be
Radon if it is finite on compact setsand inner regular with respect to compact sets.
ONDITIONALLY POSITIVE DEFINITENESS IN OPERATOR THEORY 9
Theorem , Theorem 6.2.6]) . A sequence γ = { γ n } ∞ n =0 ⊆ R is con-ditionally positive definite if and only if it has a representation of the form γ n = γ + bn + cn + Z R \{ } ( x n − − n ( x − µ ( x ) , n ∈ Z + , where b ∈ R , c ∈ R + and µ is a Radon measure on R \ { } such that Z < | x − | < ( x − d µ ( x ) < ∞ , Z | x − | > | x | n d µ ( x ) < ∞ , n ∈ Z + . For our purpose, we need the following equivalent variant of Theorem 2.2.3.
Theorem . A sequence γ = { γ n } ∞ n =0 of real numbers is conditionallypositive definite if and only if it has a representation of the form γ n = γ + bn + cn + Z R Q n ( x )d ν ( x ) , n ∈ Z + , (2.2.8) where b ∈ R , c ∈ R + and ν is a Borel measure on R such that ν ( { } ) = 0 and Z R | x | n d ν ( x ) < ∞ , n ∈ Z + . (2.2.9) Proof.
To prove the “only if” part apply Theorem 2.2.3 and define the finiteBorel measure ν on R by ν ( ∆ ) = Z ∆ ∩ ( R \{ } ) ( x − d µ ( x ) , ∆ ∈ B ( R ) . Then, by Lemma 2.2.1, the conditions (2.2.8) and (2.2.9) are satisfied (with thesame b, c ). The converse implication goes through by applying Theorem 2.2.3 tothe Radon measure µ defined by µ ( ∆ ) = Z ∆ ( x − − d ν ( x ) , ∆ ∈ B ( R \ { } ) . That the so-defined µ is a Radon measure follows from [ , Theorem 2.18]. (cid:3) Conditionally positive definite sequences of (at most) exponential growth canbe characterized as follows (below we use the convention (1.2.1)).
Theorem . Let γ = { γ n } ∞ n =0 be a sequence of real numbers. Then thefollowing conditions are equivalent :(i) γ is conditionally positive definite and there exist α, θ ∈ R + such that | γ n | α θ n , n ∈ Z + , (ii) γ is conditionally positive definite and lim sup n →∞ | γ n | /n < ∞ , (iii) there exist b ∈ R , c ∈ R + and a finite compactly supported Borel measure ν on R such that ν ( { } ) = 0 and γ n = γ + bn + cn + Z R Q n ( x )d ν ( x ) , n ∈ Z + . (2.2.10) Moreover, if (iii) holds, then the triplet ( b, c, ν ) is unique and lim sup n →∞ | γ n | /n = inf n θ ∈ R + : ∃ α ∈ R + ∀ n ∈ Z + | γ n | α θ n o , (2.2.11) c > ⇒ lim sup n →∞ | γ n | /n > , (2.2.12)supp( ν ) ⊆ h − lim sup n →∞ | γ n | /n , lim sup n →∞ | γ n | /n i , (2.2.13)sup x ∈ supp( ν ) | x | > ⇒ lim sup n →∞ | γ n | /n = sup x ∈ supp( ν ) | x | , (2.2.14)lim sup n →∞ | γ n | /n max (cid:26) , sup x ∈ supp( ν ) | x | (cid:27) . (2.2.15) Proof.
It is a matter of routine to show that the conditions (i) and (ii) areequivalent.(ii) ⇒ (iii) By Theorem 2.2.4, there exist b ∈ R , c ∈ R + and a finite Borelmeasure ν on R that satisfy the conditions (2.2.8) and (2.2.9) and the equality ν ( { } ) = 0. It follows from (2.2.7) and (2.2.8) that( △ γ ) n = Z R x n d( ν + 2 cδ )( x ) , n ∈ Z + . (2.2.16)Noting that lim sup n →∞ | ( △ γ ) n | /n lim sup n →∞ | γ n | /n (2.2.17)and using Lemma 2.1.2, we infer from (2.2.16) that( ν + 2 cδ ) (cid:16)n x ∈ R : | x | > lim sup n →∞ | γ n | /n o(cid:17) = 0 . (2.2.18)This implies (2.2.13), which gives (iii).(iii) ⇒ (ii) The conditional positive definiteness of γ follows from Theorem 2.2.4,while the inequality lim sup n →∞ | γ n | /n < ∞ can be deduced straightforwardlyfrom (2.2.1) and (2.2.10).It remains to complete the proof of the “moreover” part. The uniqueness ofthe triplet ( b, c, ν ) in (iii) follows from Lemma 2.2.2. The identity (2.2.11) is awell-known fact in analysis. The condition (2.2.12) can be deduced from (2.2.18).To prove (2.2.14), assume that R := sup x ∈ supp( ν ) | x | > ν ) = ∅ ). It isa matter of routine to deduce from (2.2.1) and (2.2.10) that there exists a constant α ∈ R + such that | γ n | α n R n , n ∈ N . This implies that lim sup n →∞ | γ n | /n R . Combined with (2.2.13), this yieldslim sup n →∞ | γ n | /n = R . Hence (2.2.14) holds. In view of (2.2.14), to prove(2.2.15), it suffices to consider the case when ν ( R \ [ − , | Q n ( x ) | (2.2.1) n − X j =0 ( n − j − | x | j n , x ∈ [ − , , n > . Combined with (2.2.10), this implies that | γ n | α · n for all n ∈ N with α = | γ | + | b | + c + ν ( R ), so lim sup n →∞ | γ n | /n
1. This completes the proof. (cid:3)
ONDITIONALLY POSITIVE DEFINITENESS IN OPERATOR THEORY 11
Definition . If γ = { γ n } ∞ n =0 is a conditionally positive definite sequencesuch that lim sup n →∞ | γ n | /n < ∞ and b , c and ν are as in the statement (iii) ofTheorem 2.2.5, we call ( b, c, ν ) the representing triplet of γ , or we simply say that( b, c, ν ) represents γ .As shown below, the converse to the implication (2.2.14) in Theorem 2.2.5 isnot true in general. Example . For θ ∈ (0 , , let γ = { γ n } ∞ n =0 be the sequence of real numbersdefined by γ n = θ n ( θ − = 1( θ − + nθ − Q n ( θ ) , n ∈ Z + . Then the sequence γ is positive definite and thus conditionally positive definite. Itsrepresenting triplet ( b, c, ν ) takes the form b = θ − , c = 0 and ν = δ θ . Moreover,we have lim sup n →∞ | γ n | /n = sup x ∈ supp( ν ) | x | = θ < , as required. ♦ Here are more examples of conditionally positive definite sequences of expo-nential growth.
Example . It is a matter of direct computation to see that if µ is a finiteBorel measure on R + , then the sequence nR [0 , x n − − x d µ ( x ) o ∞ n =0 is conditionallypositive definite and Z [0 , x n − − x d µ ( x ) (2.2.2) = − nµ ([0 , Z R + Q n ( x )d ν ( x ) , n ∈ Z + , where ν ( ∆ ) = Z ∆ ∩ [0 , (1 − x )d µ ( x ) , ∆ ∈ B ( R + ) . Similarly, if µ is a finite compactly supported Borel measure on R + , then thesequence nR (1 , ∞ ) x n − x − d µ ( x ) o ∞ n =0 is conditionally positive definite and Z (1 , ∞ ) x n − x − µ ( x ) (2.2.2) = nµ ((1 , ∞ )) + Z R + Q n ( x )d ν ( x ) , n ∈ Z + , where ν ( ∆ ) = Z ∆ ∩ (1 , ∞ ) ( x − µ ( x ) , ∆ ∈ B ( R + ) . ♦ Yet another characterization of conditionally positive definite sequences of ex-ponential growth is given below. Let us mention that in view of [ , Theorem 4.6.11]sequences γ = { γ n } ∞ n =0 ⊆ R for which △ γ is a Hausdorff moment sequence coin-cide with completely monotone sequences of order 2 introduced in [ ]. Proposition . Let γ = { γ n } ∞ n =0 be a sequence of real numbers such that lim sup n →∞ | γ n | /n < ∞ . Then the following conditions are equivalent :(i) γ is conditionally positive definite ( resp., γ is conditionally positive defi-nite with the representing triplet ( b, c, ν ) such that supp( ν ) ⊆ R + ) , (ii) △ γ is positive definite ( resp., △ γ and { ( △ γ ) n +1 } ∞ n =0 are positivedefinite ) , (iii) △ γ is a Hamburger moment sequence ( resp., △ γ is a Stieltjes momentsequence ) .Moreover, if γ is conditionally positive definite and has a representing triplet ( b, c, ν ) such that supp( ν ) ⊆ R + , then the sequence △ γ is monotonically increasing. Proof. (i) ⇒ (iii) Apply (2.2.16).(iii) ⇒ (i) Let µ be a representing measure of △ γ . Using Lemma 2.1.2 and(2.2.17) we deduce that µ is compactly supported. Note that( △ k γ ) n = ( △ k − ( △ γ )) n = Z ( x − k − x n d µ ( x ) , n ∈ Z + , k > . (2.2.19)Let ν be the finite compactly supported Borel measure on R given by ν ( ∆ ) = µ ( ∆ \ { } ) , ∆ ∈ B ( R ) . (2.2.20)Applying Newton’s binomial formula to △ (see [ , (2.2)]), we obtain γ n = n X k =0 (cid:18) nk (cid:19) ( △ k γ ) = γ + n ( γ − γ ) + Z n X k =2 (cid:18) nk (cid:19) ( x − k − d µ ( x )= γ + n ( γ − γ ) + n ( n − µ ( { } ) + Z P nk =2 (cid:0) nk (cid:1) ( x − k ( x − d ν ( x ) (2.2.2) = γ + n (cid:16) γ − γ − µ ( { } ) (cid:17) + n µ ( { } ) + Z Q n ( x )d ν ( x ) , n > . This implies that the condition (iii) of Theorem 2.2.5 holds, so γ is conditionallypositive definite. Clearly, by (2.2.20), supp( ν ) ⊆ R + if and only if supp( µ ) ⊆ R + .(ii) ⇔ (iii) Use [ , Theorem 6.2.2] and Theorem 2.1.3.Since △ γ is monotonically increasing if and only if △ γ >
0, the “moreover”part follows from (2.2.19) applied to k = 2 and (2.2.20). (cid:3) Below we give necessary and sufficient conditions for a conditionally positivedefinite sequence to have a polynomial growth of degree at most 2.
Proposition . Let γ = { γ n } ∞ n =0 be a conditionally positive definite se-quence with the representing triplet ( b, c, ν ) . Then the following conditions are equiv-alent : (i) there exists α ∈ R + such that | γ n | α · n , n ∈ N , (2.2.21)(ii) lim sup n →∞ | γ n | /n , (iii) supp( ν ) ⊆ [ − , . Moreover, (iii) implies (2.2.21) with α = | γ | + | b | + c + ν ( R ) . Proof. (i) ⇒ (ii) This implication is obvious.(ii) ⇒ (iii) It suffices to apply (2.2.13).(iii) ⇒ (i) Arguing as in the proof of (2.2.15), one can verify that the inequality(2.2.21) holds with α := | γ | + | b | + c + ν ( R ), which completes the proof. (cid:3) The above lemma enables us to prove the following.
ONDITIONALLY POSITIVE DEFINITENESS IN OPERATOR THEORY 13
Proposition . Let p be a polynomial in one indeterminate with realcoefficients. Then the following conditions are equivalent :(i) the sequence { p ( n ) } ∞ n =0 is conditionally positive definite, (ii) either deg p or deg p = 2 and the leading coefficient of p is positive. Proof.
Suppose that the sequence { p ( n ) } ∞ n =0 is conditionally positive definite.Since lim sup n →∞ | p ( n ) | /n
1, we infer from Proposition 2.2.10 that there exists α ∈ R + such that | p ( n ) | α · n for all n ∈ N . As a consequence, deg p (cid:3) Positive definite sequences of exponential growth can be characterized by meansof parameters describing conditional positive definiteness given in Theorem 2.2.5(iii).
Theorem . Let γ = { γ n } ∞ n =0 be a conditionally positive definite sequencesuch that lim sup n →∞ | γ n | /n < ∞ and let ( b, c, ν ) be the representing triplet of γ .Then the following conditions are equivalent :(i) γ is positive definite, (ii) γ is a Hamburger moment sequence, (iii) R R x − d ν ( x ) γ , b = R R x − d ν ( x ) and c = 0 .Moreover, if (ii) holds, then the sequence γ is determinate, its unique representingmeasure µ is compactly supported and the following equalities hold : µ ( ∆ ) = Z ∆ x − d ν ( x ) + (cid:16) γ − Z R x − d ν ( x ) (cid:17) δ ( ∆ ) , ∆ ∈ B ( R ) , (2.2.22) b = Z R ( x − µ ( x ) , (2.2.23) ν ( ∆ ) = Z ∆ ( x − d µ ( x ) , ∆ ∈ B ( R ) . (2.2.24) Proof.
The equivalence (i) ⇔ (ii) follows from [ , Theorem 6.2.2].(ii) ⇒ (iii) Let µ be a representing measure of γ , that is γ n = Z R x n d µ ( x ) , n ∈ Z + . (2.2.25)By Lemma 2.1.2, γ is determinate and µ is compactly supported. Note that Z R x n ( x − d µ ( x ) (2.2.25) = ( △ γ ) n (2.2.16) = Z R x n d( ν + 2 cδ )( x ) , n ∈ Z + . Since the measure ν + 2 cδ is compactly supported, we infer from Lemma 2.1.2 that Z ∆ ( x − d µ ( x ) = ( ν + 2 cδ )( ∆ ) , ∆ ∈ B ( R ) . (2.2.26)Substituting ∆ = { } into (2.2.26), we deduce that c = 0. Combined with (2.2.26),this implies (2.2.24). As a consequence of (2.2.24) and ν ( { } ) = 0, we have µ ( ∆ ) = Z ∆ x − d ν ( x ) + µ ( { } ) δ ( ∆ ) , ∆ ∈ B ( R ) . (2.2.27) If the inequality in (iii) holds, then by the Cauchy-Schwarz inequality, x − ∈ L ( ν ) . Since 1 is a common root of the polynomials X n − − n ( X − n ∈ Z + ,and ν ( { } ) = 0, it follows from Lemma 2.2.1 that γ n (2.2.10) = γ + bn + Z R x n − − n ( x − x − d ν ( x ) (2.2.24) = γ + bn + Z R (cid:16) x n − − n ( x − (cid:17) d µ ( x )= ( γ − µ ( R )) + (cid:16) b − Z R ( x − µ ( x ) (cid:17) n + Z R x n d µ ( x ) (2.2.25) = ( γ − µ ( R )) + (cid:16) b − Z R ( x − µ ( x ) (cid:17) n + γ n , n ∈ Z + . (2.2.28)Hence, we have γ = µ ( R ) (2.2.27) = Z R x − d ν ( x ) + µ ( { } ) > Z R x − d ν ( x ) , (2.2.29)and b = R R ( x − µ ( x ) , which yields the inequality in (iii) and (2.2.23). Using(2.2.23), (2.2.27) and [ , Theorem 1.29], we deduce that b = R R x − d ν ( x ). Sum-marizing, we have proved that (iii) holds. It follows from (2.2.29) that µ ( { } ) = γ − Z R x − d ν ( x ) . Combined with (2.2.27), this implies (2.2.22). This also justifies the “moreover”part.(iii) ⇒ (ii) It follows from the inequality in (iii) that the formula µ ( ∆ ) = Z ∆ x − d ν ( x ) + (cid:18) γ − Z R x − d ν ( x ) (cid:19) δ ( ∆ ) , ∆ ∈ B ( R ) , (2.2.30)defines a finite compactly supported Borel measure µ on R . Arguing as in the firstthree lines of (2.2.28) and using (2.2.30) instead of (2.2.24), we verify that (2.2.25)is satisfied. This completes the proof. (cid:3) In view of the Schur product theorem (see [ , p. 14] or [ , Theorem 7.5.3]),the product of two positive definite sequences is positive definite; this is no longertrue for conditionally positive definite sequences, e.g., the powers { n k } ∞ n =0 , k =2 , , . . . , of the conditionally positive definite sequence { n } ∞ n =0 are not conditionallypositive definite (see Proposition 2.2.11). As a consequence, if γ = { γ n } ∞ n =0 is apositive definite sequence, then the product sequence { ξ n γ n } ∞ n =0 is conditionallypositive definite for every positive definite sequence { ξ n } ∞ n =0 . Below, we show thatthe converse implication is true for sequences γ of exponential growth. Whatis more, the above equivalence remains true if the class of all positive definitesequences { ξ n } ∞ n =0 is reduced drastically to the class of the sequences of the form { θ n } ∞ n =0 , where θ ∈ R . Theorem . Suppose that { γ n } ∞ n =0 is a sequence of real numbers suchthat lim sup n →∞ | γ n | /n < ∞ . Then the following conditions are equivalent :(i) the sequence { γ n } ∞ n =0 is positive definite, (ii) the sequence { θ n γ n } ∞ n =0 is conditionally positive definite for all θ ∈ R , (iii) zero is an accumulation point of the set of all θ ∈ R \ { } for which thesequence { θ n γ n } ∞ n =0 is conditionally positive definite, ONDITIONALLY POSITIVE DEFINITENESS IN OPERATOR THEORY 15 (iv) there exists θ ∈ R \ { } such that | θ | · lim sup n →∞ | γ n | /n < and thesequence { θ n γ n } ∞ n =0 is conditionally positive definite. Proof.
The implication (i) ⇒ (ii) is a direct consequence of the Schur producttheorem. The implications (ii) ⇒ (iii) and (iii) ⇒ (iv) are obvious.(iv) ⇒ (i) Replacing { γ n } ∞ n =0 by { θ n γ n } ∞ n =0 if necessary, we can assume that { γ n } ∞ n =0 is conditionally positive definite and r := lim sup n →∞ | γ n | /n < . Then lim n →∞ γ n = 0 and consequentlylim n →∞ ( △ j γ ) n = 0 , j ∈ Z + , (2.2.31)where γ := { γ n } ∞ n =0 . Let ( b, c, ν ) be the representing triplet of γ . It follows from(2.2.13) that supp( ν ) ⊆ [ − r, r ]. Thus, by (2.2.16), we have( △ γ ) n = 2 c + Z [ − r,r ] x n d ν ( x ) , n ∈ Z + . Using (2.2.31) for j = 2 and Lebesgue’s dominated convergence theorem, we deducethat c = 0. In view of (2.2.6) and (2.2.10), we get( △ γ ) n = b + Z [ − r,r ] − x n − x d ν ( x ) , n ∈ Z + . (2.2.32)Since r <
1, we see that − x ) j ∈ L ∞ ( ν ) ⊆ L ( ν ) for all j ∈ Z + . Hence, it followsfrom (2.2.31) for j = 1, (2.2.32) and Lebesgue’s dominated convergence theoremthat b = R [ − r,r ] 1 x − d ν ( x ). According to (2.2.2) and (2.2.10), we have γ n = γ + Z [ − r,r ] (cid:18) nx − x n − − n ( x − x − (cid:19) d ν ( x )= γ + Z [ − r,r ] x n − x − d ν ( x ) , n ∈ Z + . Using (2.2.31) for j = 0 and Lebesgue’s dominated convergence theorem, we con-clude that R [ − r,r ] 1( x − d ν ( x ) = γ . Applying Theorem 2.2.12 shows that γ ispositive definite. This completes the proof. (cid:3) In this subsection we characterize condition-ally positive definite sequences γ = { γ n } ∞ n =0 of exponential growth for which thesequence of consecutive differences △ γ is either convergent (see Theorem 2.3.2) orbounded from above plus some additional constraints (see Theorem 2.3.3). As aconsequence, under slightly stronger hypotheses than those of Theorem 2.3.3, weshow that conditionally positive definite sequences γ of exponential growth withlim n →∞ ( △ γ ) n = 0 are positive definite (see Corollary 2.3.4).We begin by proving a simple lemma on backward growth estimates for powersof the difference transformation △ . Lemma . Let γ = { γ n } ∞ n =0 be a sequence of real ( resp., complex ) numbersand k ∈ N be such that sup n ∈ N ( △ k γ ) n < ∞ (cid:18) resp., sup n ∈ N | ( △ k γ ) n | < ∞ (cid:19) . (2.3.1) Then sup n ∈ N ( △ j γ ) n n k − j < ∞ (cid:18) resp., sup n ∈ N | ( △ j γ ) n | n k − j < ∞ (cid:19) , j = 0 , . . . , k. (2.3.2) Proof.
Because of the similarity of proofs, we concentrate on the real case.We use the backward induction on j . By (2.3.1), (2.3.2) holds for j = k . If the firstinequality in (2.3.2) holds for a fixed j ∈ { , . . . , k } , there exists η ∈ R + such that( △ j − γ ) n = ( △ j − γ ) + n − X m =0 ( △ j γ ) m ( △ j − γ ) + η n − X m =0 ( m + 1) k − j ( △ j − γ ) + η n k − j +1 , n ∈ N . (2.3.3)Hence the first inequality in (2.3.2) holds for j − j . (cid:3) Next, we characterize conditionally positive definite sequences γ for which thesequence △ γ is convergent. Theorem . Let γ = { γ n } ∞ n =0 be a sequence of real numbers. Then thefollowing statements are equivalent :(i) γ is conditionally positive definite andthe sequence △ γ is convergent in R , (2.3.4)(ii) there exist a finite Borel measure ν on R and d ∈ R such that (ii-a) ν ( R \ ( − , , (ii-b) − x ∈ L ( ν ) , (ii-c) γ n = γ + dn − R ( − ,
1) 1 − x n (1 − x ) d ν ( x ) for all n ∈ Z + .Moreover, the following statements are satisfied :(iii) if (i) holds and ( b, c, ν ) represents γ , then c = 0 , − x ∈ L ( ν ) and thepair ( d, ν ) with d = b + R ( − ,
1) 11 − x d ν ( x ) is a unique pair satisfying (ii) , (iv) if ν and d are as in (ii) , then d = lim n →∞ ( △ γ ) n and ( b, , ν ) represents γ with b = d − R ( − ,
1) 11 − x d ν ( x ) . Proof. (i) ⇒ (ii) It follows from footnote 2 and (2.2.13) that supp( ν ) ⊆ [ − , b, c, ν ) represents γ . By using (2.2.16), we get( △ γ ) n = 2 c + ( − n ν ( {− } ) + Z ( − , x n d ν ( x ) , n ∈ Z + . (2.3.5)It follows from (2.3.4) that lim n →∞ △ γ = 0. By Lebesgue’s dominated conver-gence theorem, the third term on the right-hand side of the equality (2.3.5) con-verges to 0. This together with (2.3.5) implies that c = 0 and ν ( {− } ) = 0, whichgives (ii-a). Now, using (2.2.6) and (2.2.10), we obtain( △ γ ) n = b + Z ( − , − x n − x d ν ( x ) + Z [0 , − x n − x d ν ( x ) , n ∈ Z + . (2.3.6)Applying Lebesgue’s dominated and monotone convergence theorems to the secondand the third terms on the right-hand side of the equality (2.3.6) respectively, we By Lemma 2.3.1 applied to k = 1 and j = 0, (2.3.4) implies that lim sup n →∞ | γ n | /n ONDITIONALLY POSITIVE DEFINITENESS IN OPERATOR THEORY 17 infer from (2.3.4) that (ii-b) holds and b = d − Z ( − , − x d ν ( x ) , (2.3.7)where d := lim n →∞ ( △ γ ) n . Using again (2.2.10), we get γ n = γ + bn + Z ( − , Q n ( x )d ν ( x ) ( ∗ ) = γ + dn + Z ( − , (cid:18) nx − x n − − n ( x − x − (cid:19) d ν ( x )= γ + dn − Z ( − , − x n (1 − x ) d ν ( x ) , n ∈ Z + , (2.3.8)where ( ∗ ) follows from (2.3.7) and (2.2.2). This implies (ii) and (iii) except for theuniqueness of ( d, ν ).(ii) ⇒ (i) Using (ii-b) and (ii-c) and arguing as in (2.3.8), we see that γ n = γ + bn + Z ( − , Q n ( x )d ν ( x ) , n ∈ Z + , (2.3.9)where b is as in (2.3.7). Hence, by Theorem 2.2.5 the sequence γ is conditionallypositive definite and lim sup n →∞ | γ n | /n < ∞ . By (ii-a), ( b, , ν ) represents γ . Itfollows from (2.2.6) and (2.3.9) that( △ γ ) n = b + Z ( − , − x n − x d ν ( x ) , n ∈ Z + . (2.3.10)Using (ii-b) and applying Lebesgue’s dominated convergence theorem to (2.3.10),we see that (2.3.4) holds and d = lim n →∞ ( △ γ ) n . Summarizing, we have provedthat (i) and (iv) hold. As a consequence, this yields the uniqueness of ( d, ν ) in (iii),which completes the proof. (cid:3) Under some additional constraints, conditionally positive definite sequences γ for which the sequence △ γ is bounded from above can be characterized as follows. Theorem . Let γ = { γ n } ∞ n =0 be a sequence of real numbers such that inf n ∈ Z + γ n > −∞ . (2.3.11) Then the following statements are equivalent :(i) γ is conditionally positive definite and sup n ∈ Z + ( △ γ ) n < ∞ , (2.3.12)supp( ν ) ⊆ R + , (2.3.13) where ν is the measure appearing in the representing triplet of γ , (ii) there exist a finite Borel measure ν on R and d ∈ R such that (ii-a) ν ( R \ [0 , , (ii-b) − x ∈ L ( ν ) , (ii-c) γ n = γ + dn − R [0 ,
1) 1 − x n (1 − x ) d ν ( x ) for all n ∈ Z + . Applying Lemma 2.3.1 to k = 1 and j = 0, we verify that (2.3.11) and (2.3.12) imply thatlim sup n →∞ | γ n | /n Moreover, the following statements are satisfied :(iii) if (i) holds and ( b, c, ν ) represents γ , then c = 0 , − x ∈ L ( ν ) and thepair ( d, ν ) with d = b + R [0 ,
1) 11 − x d ν ( x ) is a unique pair satisfying (ii) , (iv) if ν and d are as in (ii) , then d > , the sequence △ γ is monotonicallyincreasing to d and ( b, , ν ) represents γ with b = d − R [0 ,
1) 11 − x d ν ( x ) . Proof.
We begin by proving the implication (i) ⇒ (ii). Suppose (i) holds. Byfootnote 3, lim sup n →∞ | γ n | /n
1. Hence by (2.2.13) and (2.3.13), (ii-a) holds.Applying (2.2.6) and (2.2.10), we obtain( △ γ ) n = b + c (2 n + 1) + Z [0 , − x n − x d ν ( x ) , n ∈ Z + , (2.3.14)where ( b, c, ν ) represents γ . By Lebesgue’s monotone convergence theorem, thethird term on the right-hand side of the equality (2.3.14) is monotonically increasingto R [0 ,
1) 11 − x d ν ( x ). Since c >
0, we deduce from (2.3.12) and (2.3.14) that c = 0 , − x ∈ L ( ν ) (which yields (ii-b)), the sequence △ γ is monotonically increasing andconvergent in R and b = d − Z [0 , − x d ν ( x ) , (2.3.15)where d = lim n →∞ ( △ γ ) n . Using (2.2.10) and (2.3.15) and arguing as in (2.3.8),we deduce that (ii-c) holds. Since d = sup n ∈ Z + ( △ γ ) n , the telescopic argument (cf.(2.3.3)) shows that γ n γ + nd, n ∈ N . (2.3.16)Applying (2.3.11), we conclude that d >
0. This proves (ii) and (iii) except for theuniqueness of ( d, ν ).A close inspection of the proof of the implication (ii) ⇒ (i) of Theorem 2.3.2shows that (ii) implies (i) and that ( d, ν ) in (iii) is unique. By this uniqueness, thestatement (iv) follows from the proof of the implication (i) ⇒ (ii). (cid:3) Corollary . Let γ = { γ n } ∞ n =0 be a conditionally positive definite se-quence such that lim sup n →∞ | γ n | /n < ∞ and let ( b, c, ν ) be the representing tripletof γ . Suppose supp( ν ) ⊆ R + and γ n > for n large enough. Then the followingconditions are equivalent :(i) lim n →∞ ( △ γ ) n = 0 , (ii) the sequence γ is monotonically decreasing, (iii) the sequence γ is convergent in R .Moreover, if (i) holds, then γ is positive definite and γ n > for all n ∈ Z + . Proof. (i) ⇒ (ii) It follows from Theorem 2.3.3 that ν ( R \ [0 , b = R R x − d ν ( x ), c = 0 and γ n = γ − Z [0 , − x n (1 − x ) d ν ( x ) , n ∈ Z + , which yields (ii) and consequently implies that γ n > n ∈ Z + . Lebesgue’smonotone convergence theorem gives R R − x ) d ν ( x ) γ , so by Theorem 2.2.12, γ is positive definite. This proves the “moreover” part.The implications (ii) ⇒ (iii) and (iii) ⇒ (i) are obvious. (cid:3) ONDITIONALLY POSITIVE DEFINITENESS IN OPERATOR THEORY 19
3. Representations of conditionally positive definite operators3.1. Semispectral integral representations.
For the purposes of this pa-per we call an operator T ∈ B ( H ) conditionally positive definite if the sequence {k T n h k } ∞ n =0 is conditionally positive definite for every h ∈ H . Occasionally, wewill use a concise notation:( γ T,h ) n := k T n h k , n ∈ Z + , h ∈ H . (3.1.1)The class of conditionally positive definite operators is related to the class of com-plete hypercontractions of order 2 introduced by Chavan and Sholapurkar in [ ](see the paragraph preceding Proposition 3.1.4 for a more detailed discussion).Complete hypercontractions of order 2 have a semispectral integral representationthat resembles that given in Theorem 3.1.1 below. The essential difference betweenthese two approaches is that the representing semispectral measures appearing in[ ] are concentrated on the closed interval [0 , R + (let uspoint out that conditionally positive definite operators are not scalable in general,see Corollary 3.4.6). We also refer the reader to [ ] for semispectral integral rep-resentations and the corresponding dilations for completely hypercontractive andcompletely hyperexpansive operators (still on [0 , ] was an inspi-ration for the research carried out in [
20, 21 ]. Theorem . Let T ∈ B ( H ) . Then the following statements are equivalent :(i) T is conditionally positive definite, (ii) there exist operators B, C ∈ B ( H ) and a compactly supported semispectralmeasure F : B ( R + ) → B ( H ) such that B = B ∗ , C > , F ( { } ) = 0 and T ∗ n T n = I + nB + n C + Z R + Q n ( x ) F (d x ) , n ∈ Z + . (3.1.2) Moreover, if (ii) holds, then the triplet ( B, C, F ) is unique and supp( F ) ⊆ [0 , r ( T ) ] , (3.1.3) C = 0 = ⇒ r ( T ) > , (3.1.4)sup supp( F ) > ⇒ r ( T ) = sup supp( F ) . (3.1.5) Furthermore, ( h Bh, h i , h Ch, h i , h F ( · ) h, h i ) is the representing triplet of the condi-tionally positive definite sequence {k T n h k } ∞ n =0 for every h ∈ H . Proof. (i) ⇒ (ii) By Theorem 2.2.5, for every h ∈ H there exists a uniquetriplet ( b h , c h , ν h ) consisting of a real number b h , nonnegative real number c h anda finite compactly supported Borel measure ν h on R such that ν h ( { } ) = 0 and( γ T,h ) n (3.1.1) = k T n h k = k h k + b h n + c h n + Z R Q n ( x )d ν h ( x ) , n ∈ Z + . (3.1.6)First we show that supp( ν h ) ⊆ R + , h ∈ H . (3.1.7)For this, note that by (3.1.6) and (2.2.16) we have( △ γ T,h ) n = Z R x n d( ν h + 2 c h δ )( x ) , n ∈ Z + , h ∈ H . (3.1.8) It is a simple matter to verify that the following identity holds( △ γ T,h ) n +1 = ( △ γ T,T h ) n , n ∈ Z + , h ∈ H . (3.1.9)It follows from (3.1.8) and (3.1.9) that the sequences △ γ T,h and { ( △ γ T,h ) n +1 } ∞ n =0 are positive definite. Hence, by Theorem 2.1.3, △ γ T,h is a Stieltjes moment se-quence. Since the measure ν h + 2 c h δ is compactly supported, we infer from (3.1.8)and Lemma 2.1.2 that the Stieltjes moment sequence △ γ T,h is determinate (as aHamburger moment sequence). Therefore, supp( ν h + 2 c h δ ) ⊆ R + for every h ∈ H ,which implies (3.1.7).Define the functions ˆ b, ˆ c : H × H → C and ˆ ν : B ( R + ) × H × H → C byˆ b ( f, g ) = 14 X k =0 i k b f +i k g , ˆ c ( f, g ) = 14 X k =0 i k c f +i k g , ˆ ν ( ∆ ; f, g ) = 14 X k =0 i k ν f +i k g ( ∆ ) , where f, g ∈ H and ∆ ∈ B ( R + ). Clearly, ˆ ν ( · ; f, g ) is a complex measure for all f, g ∈ H . It follows from (3.1.6), (3.1.7) and the polarization formula that h T n f, T n g i = h f, g i + ˆ b ( f, g ) n + ˆ c ( f, g ) n + Z R + Q n ( x ) ˆ ν (d x ; f, g ) , n ∈ Z + , f, g ∈ H . (3.1.10)Using (3.1.10) and Lemma 2.2.2, one can verify that ˆ b is a Hermitian symmetricsesquilinear form and the functions ˆ c and ˆ ν ( ∆ ; · , -), where ∆ ∈ B ( R + ), are semi-inner products such that for all h ∈ H and ∆ ∈ B ( R + ),ˆ b ( h, h ) = b h , ˆ c ( h, h ) = c h , ˆ ν ( ∆ ; h, h ) = ν h ( ∆ ) . (3.1.11)(cf. the proofs of [ , Proposition 1] and [ , Theorem 4.2]). By (3.1.6), we haveˆ ν ( ∆ ; h, h ) + 2ˆ c ( h, h ) (3.1.11) ν h ( R ) + 2 c h (3.1.8) = ( △ γ T,h ) = h B ( T ) h, h i k B ( T ) kk h k , h ∈ H , ∆ ∈ B ( R + ) . This implies that the sesquilinear forms ˆ c and ˆ ν ( ∆ ; · , -), where ∆ ∈ B ( R + ), arebounded. Hence, there exist C, F ( ∆ ) ∈ B ( H ) + , where ∆ ∈ B ( R + ), such that h Ch, h i = ˆ c ( h, h ) (3.1.11) = c h , h ∈ H , (3.1.12) h F ( ∆ ) h, h i = ˆ ν ( ∆ ; h, h ) (3.1.11) = ν h ( ∆ ) , ∆ ∈ B ( R + ) , h ∈ H . (3.1.13)In view of (3.1.13), F is a Borel semispectral measure on R + .Now we show that the so-constructed F satisfies (3.1.3) and (3.1.5). It followsfrom Gelfand’s formula for spectral radius thatlim sup n →∞ k T n h k /n r ( T ) , h ∈ H . (3.1.14)This together with (3.1.6), (3.1.13) and Theorem 2.2.5 applied to γ T,h yields D F (cid:16)(cid:0) r ( T ) , ∞ (cid:1)(cid:17) h, h E D F (cid:16)(cid:0) lim sup n →∞ k T n h k /n , ∞ (cid:1)(cid:17) h, h E (2.2.13) = 0 , h ∈ H , ONDITIONALLY POSITIVE DEFINITENESS IN OPERATOR THEORY 21 which, when combined with (3.1.7), implies (3.1.3). Hence, we havesup supp( F ) r ( T ) . (3.1.15)Observing that supp( h F ( · ) h, h i ) ⊆ supp( F ) , h ∈ H , we obtain lim sup n →∞ k T n h k /n (2.2.15) max (cid:26) , sup supp( h F ( · ) h, h i ) (cid:27) max (cid:8) , sup supp( F ) (cid:9) , h ∈ H . It follows from [ , Corollary 3] that r ( T ) max (cid:26) , sup supp( F ) (cid:27) , which to-gether with (3.1.15) gives (3.1.5).Our next goal is to construct the operator B . By (3.1.6) and (3.1.12), we have k T h k − k h k = ( △ γ T,h ) = ˆ b ( h, h ) + h Ch, h i , h ∈ H . As a consequence, ˆ b is a bounded Hermitian symmetric sesquilinear form. Thisimplies that the exists a selfadjoint operator B ∈ B ( H ) such that h Bh, h i = ˆ b ( h, h ) (3.1.11) = b h , h ∈ H . (3.1.16)Combining (3.1.6) with (3.1.12), (3.1.13) and (3.1.16) gives (ii).(ii) ⇒ (i) This implication is a direct consequence of Theorem 2.2.5 applied tothe sequences γ T,h , h ∈ H .It remains to justify the “moreover” part. Suppose (ii) holds. The uniquenessof the triplet ( B, C, F ) follows from Theorem 2.2.5. The assertions (3.1.3) and(3.1.5) were proved above. To show (3.1.4), assume that C = 0. Then the set U := { h ∈ H : h Ch, h i > } is nonempty. By the “moreover” part of Theorem 2.2.5and (3.1.14), we have r ( T ) > lim sup n →∞ k T n h k /n > , h ∈ U, which implies (3.1.4). The last statement of the theorem is easily seen to be true.This completes the proof. (cid:3) The following definition is an operator counterpart of Definition 2.2.6.
Definition . If T ∈ B ( H ) is a conditionally positive definite operatorand B , C and F are as in the statement (ii) of Theorem 3.1.1, we call ( B, C, F ) the representing triplet of T , or we simply say that ( B, C, F ) represents T . Remark . Note that if (
B, C, F ) represents a conditionally positive defi-nite operator T on H 6 = { } and B >
0, then by (3.1.2), T ∗ n T n > I for every n ∈ N which together with Gelfand’s formula for spectral radius yields r ( T ) > ♦ Proposition 3.1.4 below which gives characterizations of conditionally positivedefinite operators is closely related to Proposition 2.2.9 (see also Theorem 3.3.1 foran alternative approach). The most important fact we need in its proof is thata sequence { γ n } ∞ n =0 ⊆ B ( H ) of exponential growth is a Hamburger moment se-quence (that is, (2.1.2) holds for some semispectral measure µ : B ( R ) → B ( H ))if and only if {h γ n h, h i} ∞ n =0 is a Hamburger moment sequence for all h ∈ H (see [ , Theorem 2]). Similar assertions are true for Stieltjes and Hasudorff operatormoment sequences. In view of [ ], operators T ∈ B ( H ) for which the sequence { T ∗ n B ( T ) T n } ∞ n =0 is a Hausdorff moment sequence coincide with complete hyper-contractions of order 2 introduced in [ ]. On the other hand, by [ , Corollary](see also Theorem 1.1.1), an operator T ∈ B ( H ) is subnormal if and only if the se-quence { T ∗ n T n } ∞ n =0 is a Stieltjes moment sequence. We refer the reader to [
70, 13 ]for necessary definitions and facts related to the aforesaid operator moment prob-lems.
Proposition . For T ∈ B ( H ) , the following conditions are equivalent :(i) T is conditionally positive definite, (ii) { T ∗ n B ( T ) T n } ∞ n =0 is positive definite, (iii) { T ∗ n B ( T ) T n } ∞ n =0 is a Stieltjes moment sequence.Moreover, if T is conditionally positive definite, then { T ∗ ( n +1) T n +1 − T ∗ n T n } ∞ n =0 is monotonically increasing and inf n ∈ Z + ( k T n +1 h k − k T n h k ) = −h B ( T ) h, h i = h Bh, h i + h Ch, h i , h ∈ H , where B and C are as in Theorem . Proof.
Clearly, the sequence { T ∗ n B ( T ) T n } ∞ n =0 is of exponential growth and( △ γ T,h ) n = h T ∗ n B ( T ) T n h, h i for all n ∈ Z + and h ∈ H . This implies that { T ∗ n B ( T ) T n } ∞ n =0 is positive definite (resp., a Stieltjes moment sequence) if andonly if △ γ T,h is positive definite (resp., a Stieltjes moment sequence) for every h ∈ H . Applying Proposition 2.2.9 to the sequences γ T,h and using Theorem 3.1.1,we deduce that the conditions (i)-(iii) are equivalent. The “moreover” part is adirect consequence of the corresponding part of Proposition 2.2.9, Theorem 3.1.1and (3.1.2) applied to n = 1. (cid:3) The next result can be thought of as an operator counterpart of Theorem 2.3.3.Note that if an operator T ∈ B ( H ) is conditionally positive definite, then by Propo-sition 3.1.4, the sequence { T ∗ ( n +1) T n +1 − T ∗ n T n } ∞ n =0 is monotonically increasing,which implies that “sup n ∈ Z + ” in (3.1.17) can be replaced by “lim n →∞ ” (in theextended real line). Hence, if (3.1.17) holds, then the sequence { T ∗ ( n +1) T n +1 − T ∗ n T n } ∞ n =0 is convergent in wot , say to D ∈ B ( H ). It is worth mentioning that inview of Remark 4.3.3a) and [ , Proposition 8] (see also [ , Lemma 6.1(ii)]), thereare conditionally positive definite weighted shift operators T such that h Dh, h i = sup n ∈ Z + ( k T n +1 h k − k T n h k ) = −h B ( T ) h, h i > , h ∈ H \ { } . In turn, there are conditionally positive definite operators T for which the onlyvector h satisfying (3.1.17) is the zero vector (see Remark 4.3.3c)). Theorem . Let T ∈ B ( H ) . Then the following are equivalent :(i) T is conditionally positive definite and sup n ∈ Z + ( k T n +1 h k − k T n h k ) < ∞ , h ∈ H , (3.1.17)(ii) there exist a semispectral measure F : B ( R + ) → B ( H ) and a selfadjointoperator D ∈ B ( H ) such that (ii-a) F ([1 , ∞ )) = 0 , (ii-b) − x ∈ L ( F ) , ONDITIONALLY POSITIVE DEFINITENESS IN OPERATOR THEORY 23 (ii-c) T ∗ n T n = I + nD − R [0 ,
1) 1 − x n (1 − x ) F (d x ) for all n ∈ Z + .Moreover, the following statements are satisfied :(iii) if (i) holds and ( B, C, F ) represents T , then C = 0 , − x ∈ L ( F ) and thepair ( D, F ) with D = B + R [0 ,
1) 11 − x F (d x ) is a unique pair satisfying (ii) , (iv) if F and D are as in (ii) , then D > , { T ∗ ( n +1) T n +1 − T ∗ n T n } ∞ n =0 converg-es in wot to D and ( B, , F ) represents T with B = D − R [0 ,
1) 11 − x F (d x ) . Proof.
Using Theorem 2.3.3 (together with its “moreover” part) and Theo-rem 3.1.1 (together with its “furthermore” part), we deduce that the statements (i)and (ii) are equivalent and the statements (iii) and (iv) hold. (cid:3)
The simple argument given below shows that the condition (3.1.17) is strongenough to guarantee that r ( T ) Proposition . If an operator T ∈ B ( H ) satisfies the condition (3.1.17) ,then α T ( h ) := sup n ∈ Z + ( k T n +1 h k − k T n h k ) > for all h ∈ H and r ( T ) . Proof.
Using the telescopic argument (cf. (2.3.16)) yields k T n h k k h k + nα T ( h ) , n ∈ N , h ∈ H . Hence, α T ( h ) > n →∞ k T n h k /n h ∈ H . Applying [ ,Corollary 3], we conclude that r ( T ) (cid:3) We show below that if the operator D in Theorem 3.1.5 is nonzero, then thespectral radius of T is equal to 1. The case D = 0 is discussed in Theorem 3.4.4. Proposition . Let T ∈ B ( H ) be a conditionally positive definite operatorsatisfying (3.1.17) . Suppose that − x ) ∈ L ( F ) and D = 0 , where F is as in Theo-rem and D := (wot) lim n →∞ ( T ∗ ( n +1) T n +1 − T ∗ n T n ) . Then r ( T ) = 1 . Proof.
By the statements (iii) and (iv) of Theorem 3.1.5, the pair (
D, F )satisfies the condition (ii) of this theorem and D >
0. Hence h Dh , h i > h ∈ H . In view of Lebesgue’s monotone convergence theorem, the sequence { R [0 ,
1) 1 − x n (1 − x ) h F (d x ) h , h i} ∞ n =0 converges to R [0 ,
1) 1(1 − x ) h F (d x ) h , h i . Because thelast integral is finite, we infer from the equality (ii-c) of Theorem 3.1.5 that thereexists n ∈ N such that k T n h k n = k h k − R [0 ,
1) 1 − x n (1 − x ) h F (d x ) h , h i n + h Dh , h i > h Dh , h i , n > n . Combined with Gelfand’s formula for spectral radius, this implies that r ( T ) > lim sup n →∞ k T n h k /n n / n > . Therefore applying Proposition 3.1.6 yields r ( T ) = 1. (cid:3) It follows from Theorem 3.1.5(ii-c) that if T ∈ B ( H ) is a conditionally pos-itive definite operator satisfying (3.1.17), then there exists α ∈ R + such that k T n k α √ n for all n ∈ N . Proposition 3.1.8 below shows that the powers ofa conditionally positive definite operator with spectral radius less than or equal to1 have polynomial growth of degree at most 1. According to Proposition 4.3.1 and Remark 4.3.3d) there are conditionally positive definite operators having exactlypolynomial growth of degree 1. Observe that a subnormal operator of polynomialgrowth of arbitrary degree, being normaloid (see (1.2.2)) is a contraction, thatis, it is either a zero operator or it has exactly polynomial growth of degree 0.Proposition 3.1.8 can be deduced from Proposition 2.2.10 and Gelfand’s formulafor spectral radius.
Proposition . Let T ∈ B ( H ) be a conditionally positive definite operatorwith the representing triplet ( B, C, F ) . Then the following conditions are equivalent :(i) there exists α ∈ R + such that k T n k α · n, n ∈ N , (3.1.18)(ii) r ( T ) , (iii) supp( F ) ⊆ [0 , . Moreover, (iii) implies (3.1.18) with α = p k B k + k C k + k F ([0 , k . First, we adapt Agler’s hereditary func-tional calculus [
1, 47, 26 ] to our needs. For T ∈ B ( H ), we set p h T i = X i > α i T ∗ i T i for p = X i > α i X i ∈ C [ X ] . (3.2.1)In particular, we have (see (1.2.3)) B m ( T ) = (1 − X ) m h T i , m ∈ Z + . (3.2.2)The map C [ X ] ∋ p p h T i ∈ B ( H ) is linear but in general not multiplicative (e.g.,if T ∈ B ( H ) is a nilpotent operator with index of nilpotency 2 and p = X −
1, then p h T i = ( p ) h T i ). However, it has the following property. The map p p h T i is a unique linear map from C [ X ] to B ( H ) suchthat X h T i = I and ( Xp ) h T i = T ∗ p h T i T for all p ∈ C [ X ] . (3.2.3)There is another way of defining p h T i . Namely, let us consider the elementaryoperator ∇ T : B ( H ) → B ( H ) defined by ∇ T ( A ) = T ∗ AT for A ∈ B ( H ). It is theneasily seen that p h T i = p ( ∇ T )( I ) for any p ∈ C [ X ] and by (3.2.3), p ( ∇ T )( q h T i ) = (( pq )( ∇ T )) ( I ) = ( pq ) h T i , p, q ∈ C [ X ] . (3.2.4)Although the map C [ X ] ∋ p p h T i ∈ B ( H ) is not multiplicative, it does have aproperty that resembles multiplicativity. Lemma . Let T ∈ B ( H ) and q ∈ C [ X ] . Then the set I = { q ∈ C [ X ] : ( q q ) h T i = 0 } is a principal ideal in C [ X ] , that is, I = { pw : p ∈ C [ X ] } for some w ∈ C [ X ] .Moreover, if q = q ∗ , then w can be chosen to satisfy w ∗ = w . Proof.
First note that I is an ideal. Indeed, if q ∈ I and p ∈ C [ X ], then0 = p ( ∇ T ) (cid:16)(cid:0) q q (cid:1) h T i (cid:17) (3.2.4) = ( q pq ) h T i . By [ , Theorem III.3.9], I is a principal ideal in C [ X ]. That w can be chosen tosatisfy w ∗ = w , follows from the fact that ( p h T i ) ∗ = ( p ∗ h T i ) for all p ∈ C [ X ]. (cid:3) ONDITIONALLY POSITIVE DEFINITENESS IN OPERATOR THEORY 25
Remark . It follows from (3.2.2) and Lemma 3.2.1 that if T ∈ B ( H ) isan m -isometry, that is (1 − X ) m h T i = B m ( T ) = 0, then ((1 − X ) m q ) h T i = 0 forevery q ∈ C [ X ]; in particular, B k ( T ) = (1 − X ) k h T i = 0 for all k > m , whichmeans that T is a k -isometry for all k > m (see [ , p. 389, line 6]). ♦ Given a ∈ C , we define the linear transformation D a : C [ X ] → C [ X ] by D a p = p − p ( a ) X − a , p ∈ C [ X ] . Using the Taylor’s series expansion about the point a , it is easily seen that thetransformation D a is well defined and ( p ′ ( a ) stands for the derivative of p at a ) D a p = p − p ( a ) − p ′ ( a )( X − a )( X − a ) , a ∈ C , p ∈ C [ X ] . (3.2.5)It is a simple matter to verify that for each p ∈ C [ X ], D na p = 0 whenever n > deg p .The following lemma will be used in some proofs of subsequent results. Lemma . The following assertions hold :(a) if K is a Hilbert space, S ∈ B ( K ) + , E is the spectral measure of S and M is a vector subspace of K , then _ n E ( ∆ ) M : ∆ ∈ B ( R + ) o = _ { S n M : n ∈ Z + } , (3.2.6)(b) if for i = 1 , , ( K i , R i , S i ) consists of a Hilbert space K i and operators R i ∈ B ( H , K i ) and S i ∈ B ( K i ) + such that K i = W { S ni R ( R i ) : n ∈ Z + } and R ∗ S n R = R ∗ S n R for all n ∈ Z + , then there exists a ( unique ) unitaryisomorphism U ∈ B ( K , K ) such that U R = R and U S = S U ; inparticular, σ ( S ) = σ ( S ) and k S k = k S k . Proof. (a) Since S is bounded, E ( R + \ [0 , r ]) = 0, where r := k S k . Take avector g ∈ K . Then g is orthogonal to the right-hand side of (3.2.6) if and only if0 = h S n h, g i = Z [0 ,r ] x n h E (d x ) h, g i , n ∈ Z + , h ∈ M . (3.2.7)It follows from the Weierstrass approximation theorem and the uniqueness part ofthe Riesz representation theorem (see [ , Theorem 6.19]) that (3.2.7) holds if andonly if h E ( ∆ ) h, g i = 0 for all ∆ ∈ B ( R + ) and h ∈ M , or equivalently if and onlyif g is orthogonal to the left-hand side of (3.2.6). This implies (3.2.6).(b) It is easily seen that there exists a unique unitary isomorphism U ∈ B ( K , K ) such that U S n R h = S n R h for all h ∈ H and n ∈ Z + . It is a matter ofroutine to verify that U has the desired properties. This completes the proof. (cid:3) For the reader’s convenience, we recall a version of the Naimark dilation theo-rem needed in this paper.
Theorem , Theorem 6.4]) . If M : B ( R + ) → B ( H ) is a semispectralmeasure, then there exist a Hilbert space K , an operator R ∈ B ( H , K ) and a spectralmeasure E : B ( R + ) → B ( K ) such that M ( ∆ ) = R ∗ E ( ∆ ) R, ∆ ∈ B ( R + ) , (3.2.8) K = _ (cid:8) E ( ∆ ) R ( R ) : ∆ ∈ B ( R + ) (cid:9) . (3.2.9) We are now ready to give a dilation representation for conditionally positivedefinite operators and relate their spectral radii to the norms of positive operatorsappearing in this representation. Dilation representations for complete hypercon-tractions and complete hyperexpansions were given in [ ] and afterwards general-ized to the case of complete hypercontractions of finite order in [ ]. All aforesaidrepresentations were built over the closed interval [0 , , Theorem 4.20]).Below we use the convention (1.2.1). Theorem . Let T ∈ B ( H ) . Then the following conditions are equivalent :(i) T is conditionally positive definite, (ii) there exists a semispectral measure M : B ( R + ) → B ( H ) with compactsupport such that p h T i = p (1) I − p ′ (1) B ( T ) + Z R + ( D p )( x ) M (d x ) , p ∈ C [ X ] , (3.2.10)(iii) there exist a Hilbert space K , R ∈ B ( H , K ) and S ∈ B ( K ) + such that p h T i = p (1) I − p ′ (1) B ( T ) + R ∗ ( D p )( S ) R, p ∈ C [ X ] , (3.2.11)(iv) there exist a Hilbert space K , R ∈ B ( H , K ) and S ∈ B ( K ) + such that (3.2.11) holds and K = _ { S n R ( R ) : n ∈ Z + } . (3.2.12) Moreover, if any of the conditions (i)-(iv) holds, then (a) the semispectral measure M in (ii) is unique, (b) if ( B, C, F ) represents T , then B + C = − B ( T ) , C = M ( { } ) and F ( ∆ ) = (1 − χ ∆ (1)) M ( ∆ ) , ∆ ∈ B ( R + ) , (3.2.13)(c) if ( K , R, S ) is as in (iv) , then σ ( S ) = supp( M ) and k S k = max (cid:8) , sup supp( M ) (cid:9) r ( T ) . (3.2.14) Proof. (i) ⇒ (ii) By Theorem 3.1.1, T has a representing triplet ( B, C, F ).Define the compactly supported semispectral measure M : B ( R + ) → B ( H ) by M ( ∆ ) = F ( ∆ ) + 2 χ ∆ (1) C, ∆ ∈ B ( R + ) . Using (3.1.2) and the fact that Q n (1) = n ( n − (see (2.2.1)), we deduce that T ∗ n T n = I + n ( B + C ) + Z R + Q n ( x ) M (d x ) , n ∈ Z + . (3.2.15)Since Q = 0, substituting n = 1 into (3.2.15) yields B + C = − B ( T ). Hence T ∗ n T n = I − n B ( T ) + Z R + Q n ( x ) M (d x ) , n ∈ Z + . (3.2.16)Suppose p ∈ C [ X ] is of the form p = P n > α n X n , where α n ∈ C . Multiplying(3.2.16) by α n and summing with respect to n , gives p h T i (3.2.1) = p (1) I − p ′ (1) B ( T ) + Z R + X n > α n Q n ( x ) M (d x ) . (3.2.17) ONDITIONALLY POSITIVE DEFINITENESS IN OPERATOR THEORY 27
Notice that X n > α n Q n ( x ) (2.2.2) = X n > α n x n − − n ( x − x − = ( D p )( x ) , x ∈ R \ { } . (3.2.18)Combining (3.2.17) with (3.2.18) gives (3.2.10).(ii) ⇒ (i) Substituting the polynomial p = X n into (3.2.10) and using (2.2.2),(3.2.5), (3.2.1) and the fact that Q n (1) = n ( n − , we get T ∗ n T n = I − n B ( T ) + Z R + Q n ( x ) M (d x )= I − n B ( T ) + Q n (1) M ( { } ) + Z R + Q n ( x ) F (d x )= I − n (cid:18) B ( T ) + 12 M ( { } ) (cid:19) + n M ( { } ) + Z R + Q n ( x ) F (d x ) , n ∈ Z + , (3.2.19)where F : B ( R + ) → B ( H ) is the compactly supported semispectral measure givenby (3.2.13). Hence, by Theorem 3.1.1, T is conditionally positive definite. What ismore, using (3.2.13), (3.2.19) and the uniqueness of representing triplets, we easilyverify that (a) and (b) hold.(ii) ⇒ (iv) By Theorem 3.2.4, there exists a triplet ( K , R, E ) satisfying (3.2.8)and (3.2.9). Notice that the measure E is compactly supported. This is a directconsequence of the identity supp( M ) = supp( E ), which follows from (3.2.8) and(3.2.9) (see the proof of [ , Theorem 4.4]). Set S = R R + xE (d x ). Since E iscompactly supported in R + , the operator S is bounded and positive (see [ , The-orem 5.9]). Applying the Stone-von Neumann functional calculus (cf. [
12, 56 ] and[ ]), we deduce from (3.2.10) that the triplet ( K , R, S ) satisfies (3.2.11). Using(3.2.9) and Lemma 3.2.3(a), we get (3.2.12), which yields (iv).(iv) ⇒ (iii) This is obvious.(iii) ⇒ (ii) Applying the Stone-von Neumann functional calculus to (3.2.11)yields (ii) with the semispectral measure M defined by (3.2.8), where E is thespectral measure of S .It remains to prove (c). Suppose that ( K , R, S ) is as in (iv). If K = { } , thenby (3.2.10), (3.2.11) and (a), we deduce that M = 0 which gives (3.2.14). Thereforewe can assume that K 6 = { } . According to the proof of the implication (iv) ⇒ (ii),we see that the semispectral measure M is given by (3.2.8), where E is the spectralmeasure of S . In view of (3.2.12) and Lemma 3.2.3(a), ( K , R, E ) satisfies (3.2.9),and so M = 0. As mentioned above, supp( M ) = supp( E ). Combined with [ ,Theorem 5.9, Proposition 5.10], this implies that σ ( S ) = supp( M ) and k S k = sup σ ( S ) = sup supp( M ).Thus it suffices to show that sup supp( M ) r ( T ) . Let ( B, C, F ) be the repre-senting triplet of T . Set ϑ = sup supp( F ). Since supp( F ) is compact, we see that ϑ ∈ {−∞} ∪ R + . We now consider three possible cases that are logically disjoint.The possibility that ϑ = −∞ (equivalently, supp( F ) = ∅ ) may happen only in Case2 (see Corollary 3.2.7 and Remark 4.3.3d)). Case 1. ϑ > r ( T ) = sup supp( F ) (b) = sup supp( M ) . Case 2. ϑ < C = 0.According to (3.1.4), r ( T ) >
1. Since ϑ <
1, we obtainsup supp( M ) (b) = 1 r ( T ) . Case 3. ϑ < C = 0.First observe that by (b), ϑ = sup supp( M ). It follows from (3.1.3) that ϑ r ( T ) , hence sup supp( M ) r ( T ) . This completes the proof. (cid:3) Corollary . Suppose T ∈ B ( H ) is a conditionally positive definite op-erator and ( K , R, S ) is as in Theorem . If is an accumulation point of σ ( S ) ∩ (0 , or if σ ( S ) ∩ (1 , ∞ ) = ∅ , then r ( T ) = k S k . Proof.
Let (
B, C, F ) be the representing triplet of T and let M be as inTheorem 3.2.5(ii). Suppose first that 1 is an accumulation point of σ ( S ) ∩ (0 , M ) = supp( F )and 1 ∈ supp( F ), so by the second equality in (3.2.14), we have k S k = sup supp( F ) > . In turn, if σ ( S ) ∩ (1 , ∞ ) = ∅ , then again by the first equality in (3.2.14) and (3.2.13),1 < sup σ ( S ) = sup supp( M ) = sup supp( F ) . In both cases, an application of (3.1.5) and sup σ ( S ) = k S k yields r ( T ) = k S k . (cid:3) The next corollary enables as to determine the mass of the measure M at thepoint 0 provided the conditionally positive definite operator has the spectral radiusless than or equal to 1. Corollary . Suppose T ∈ B ( H ) is a conditionally positive definite op-erator and M is as in Theorem . Then B m ( T ) = Z R + (1 − x ) m − M (d x ) , m ≥ . (3.2.20) In particular, the following assertions hold :(i) B k ( T ) > for all k ∈ Z + , (ii) if r ( T ) , then B m ( T ) > for all m ∈ Z + \ { } , (iii) if r ( T ) , then the sequence { B m ( T ) } ∞ m =2 is monotonically decreasingand convergent to M ( { } ) in the strong operator topology. Proof.
Fix an integer m > p = (1 − X ) m . Then by (3.2.5) we have D p = (1 − X ) m − . Applying (3.2.2) and Theorem 3.2.5(ii), we get (3.2.20). Theassertion (i) is immediate from (3.2.20), while the assertions (ii) and (iii) can bededuced from (3.1.3), (3.2.13) and Lebesgue’s dominated convergence theorem. (cid:3) Concluding this subsection, we make a few remarks related to Theorem 3.2.5and Corollary 3.2.7.
ONDITIONALLY POSITIVE DEFINITENESS IN OPERATOR THEORY 29
Remark . a) Let us begin by discussing in more detail the relationshipbetween r ( T ) and ϑ = sup supp( F ), where T ∈ B ( H ) is a conditionally positivedefinite operator and ( B, C, F ) represents T . As in the proof of Theorem 3.2.5(c),we consider three cases. If ϑ >
1, then by (3.1.5), 1 ϑ = r ( T ) . If ϑ < C = 0, then by (3.1.4), ϑ < r ( T ) .Suppose now that ϑ < C = 0. First, we consider the subcase when D := B + R R + − x F (d x ) = 0. Then, there exists h ∈ H such that η ( h ) := h Dh , h i 6 = 0. According to (3.1.2), we have k T n h k = n (cid:18) k h k n + h Bh , h i + Z R + Q n ( x ) n h F (d x ) h , h i (cid:19) , n ∈ N . (3.2.21)By assumption that ϑ <
1, we infer from (2.2.4), (2.2.5) and Lebesgue’s monotoneconvergence theorem that h Bh , h i + Z R + Q n ( x ) n h F (d x ) h , h i −→ η ( h ) as n → ∞ .Since η ( h ) = 0, we deduce from (3.2.21) that η ( h ) > r ( T ) > lim sup n →∞ k T n h k /n > > ϑ. It remains to consider the subcase when D = 0. Then by (1.2.2), (3.1.3) andTheorem 3.4.4(v), T is subnormal and ϑ r ( T ) = k T k σ ( S ) does notdepend on a triplet ( K , R, S ) satisfying (3.2.11) and (3.2.12). This fact can be alsodeduced from Lemma 3.2.3(b) by applying (3.2.11) and (3.2.5) to the polynomials p = ( X − X n , where n ∈ Z + .c) Concerning (3.2.14), observe that if T ∈ B ( H ) is a 2-isometry and H 6 = { } , then T is conditionally positive definite and 1 = r ( T ) > k S k = 0 (useProposition 4.3.1, (3.2.14) and [ , Lemma 1.21]).d) Regarding Corollary 3.2.7 (see also Proposition 3.3.5), it is worth recalling aresult due to Agler saying that an operator T ∈ B ( H ) is a subnormal contractionif and only if B m ( T ) > m ∈ Z + (see [ , Theorem 3.1]). Recently Guhas shown that B m ( T ) > B m − ( T ) > m (see [ , Theorem 2.5]). It turns out that there are non-subnormal conditionallypositive definite operators T with r ( T ) = 1 , so by Corollary 3.2.7(ii) for such T ’s, B m ( T ) > m ∈ Z + \ { } (see e.g., Example 4.3.6; cf. also [ ,Section 9]). ♦ First, followingProposition 3.1.4, we simplify the previous representations of conditionally posi-tive definite operators.
Theorem . For T ∈ B ( H ) , the following conditions are equivalent :(i) T is conditionally positive definite, (ii) there exists a semispectral measure M : B ( R + ) → B ( H ) with compactsupport such that (( X − q ) h T i = Z R + q ( x ) M (d x ) , q ∈ C [ X ] , (3.3.1) (ii ′ ) there exist a Hilbert space K , R ∈ B ( H , K ) and S ∈ B ( K ) + such that (( X − q ) h T i = R ∗ q ( S ) R, q ∈ C [ X ] , (3.3.2)(iii) there exists a semispectral measure M : B ( R + ) → B ( H ) with compactsupport such that T ∗ n B ( T ) T n = Z R + x n M (d x ) , n ∈ Z + , (3.3.3)(iii ′ ) there exist a Hilbert space K , R ∈ B ( H , K ) and S ∈ B ( K ) + such that T ∗ n B ( T ) T n = R ∗ S n R, n ∈ Z + . (3.3.4) Moreover, the measures in (ii) and (iii) are unique and coincide with that in Theo-rem the triplets ( K , R, S ) in (ii ′ ) and (iii ′ ) can be chosen to satisfy (3.2.12) . Proof. (i) ⇒ (ii ′ ) Applying the implication (i) ⇒ (iv) of Theorem 3.2.5 to thepolynomial p = ( X − q and using (3.2.5), we get a triplet ( K , R, S ) satisfying(3.3.2) and (3.2.12).(ii ′ ) ⇒ (i) It follows from (3.2.5) that p = p (1) + p ′ (1)( X −
1) + ( X − D p, p ∈ C [ X ] . Since the mapping p p h T i is linear, we obtain p h T i = p (1) I − p ′ (1) B ( T ) + (cid:16) ( X − D p (cid:17) h T i (3.3.2) = p (1) I − p ′ (1) B ( T ) + R ∗ ( D p )( S ) R, p ∈ C [ X ] , which means that (3.2.11) holds. Applying Theorem 3.2.5 gives (i).(ii ′ ) ⇔ (iii ′ ) One can easily check that these two conditions are equivalent withthe same triplet ( K , R, S ) (use (3.2.2) and (3.2.4)). This together with the firstparagraph of this proof justifies the second statement of the “moreover” part.Arguing as in the proof of the equivalence (ii) ⇔ (iv) of Theorem 3.2.5, we de-duce that the equivalences (ii) ⇔ (ii ′ ) and (iii) ⇔ (iii ′ ) hold. The first statement of the“moreover” part can be inferred from Theorem 3.2.5(a) by observing that the con-ditions (3.2.10), (3.3.1) and (3.3.3) are equivalent (cf. the proof of the equivalence(i) ⇔ (ii ′ )). This completes the proof. (cid:3) Many classes of operators are closed under the operation of taking powers.Among them are the classes of normaloid, subnormal, k -isometric, k -expansive,completely hyperexpansive and alternatingly hyperexpansive operators (see [ , p.99], [ , Theorem 2.3] and [ , Theorem 2.3]). On the other hand, the class ofhyponormal operators does not share this property (see [ , Problem 209]). As thefirst application of Theorem 3.3.1, we show that the class of conditionally positivedefinite operators does share this property. We also describe the semispectral andthe dilation representations for powers of conditionally positive definite operators. Proposition . Suppose that T ∈ B ( H ) is a conditionally positive definiteoperator and i ∈ N \ { } . Then (i) T i is conditionally positive definite, (ii) if M and M i are semispectral measures that correspond respectively to T and T i via Theorem , then M i ( ∆ ) = ˜ M i ( ψ − i ( ∆ )) , ∆ ∈ B ( R + ) , (3.3.5) ONDITIONALLY POSITIVE DEFINITENESS IN OPERATOR THEORY 31 where ˜ M i : B ( R + ) → B ( H ) is the semispectral measure defined by ˜ M i ( ∆ ) = Z ∆ (1 + x + . . . + x i − ) M (d x ) , ∆ ∈ B ( R + ) , (3.3.6) and ψ i : R + → R + is given by ψ i ( x ) = x i for x ∈ R + , (iii) the representing triplet ( B i , C i , F i ) of T i can be described by applying The-orem to M i in place of M , (iv) if ( K , R, S ) is as in Theorem , then the triplet ( K , R i , S i ) with R i := ( I + S + . . . + S i − ) R, (3.3.7) corresponds to T i via Theorem . Proof. (i)&(ii) First, it is easily seen that B ( T i ) = (1 − X i ) h T i . (3.3.8)Let M be as in Theorem 3.2.5(ii). By the “moreover” part of Theorem 3.3.1, M satisfies (3.3.1). Clearly, the set functions ˜ M i and M i defined by (3.3.6) and (3.3.5),respectively, are semispectral measures that are compactly supported. Applying(1.2.4) and the measure transport theorem, we get( T i ) ∗ n B ( T i )( T i ) n (3.3.8) = ( ∇ T ) in (cid:0) (1 − X i ) h T i (cid:1) (3.2.4) = (cid:0) X in (1 − X i ) (cid:1) h T i = (cid:0) X in (1 + X + . . . + X i − ) (1 − X ) (cid:1) h T i (3.3.1) = Z R + x in (1 + x + . . . + x i − ) M (d x )= Z R + ψ i ( x ) n ˜ M i (d x )= Z R + t n M i (d t ) , n ∈ Z + . Using Theorem 3.3.1(iii) and the “moreover” part of this theorem, we see that (i)and (ii) hold.(iii) Obvious.(iv) Let ( K , R, S ) be as in Theorem 3.2.5(iv). Denote by E S and E S i thespectral measures of S and S i , respectively. In view of [ , Theorem 6.6.4], we have E S i ( ∆ ) = E S ( ψ − i ( ∆ )) , ∆ ∈ B ( R + ) . (3.3.9)According to the proof of the implication (iii) ⇒ (ii) of Theorem 3.2.5, M ( ∆ ) = R ∗ E S ( ∆ ) R, ∆ ∈ B ( R + ) . (3.3.10)It follows from (3.2.12) and Lemma 3.2.3(a) that K = _ (cid:8) E S ( ∆ ) R ( R ) : ∆ ∈ B ( R + ) (cid:9) . (3.3.11)Define the function ζ i : R + → R + by ζ i ( x ) = 1 + x + . . . + x i − for x ∈ R + . Using(3.3.5) and (3.3.6) and applying the Stone-von Neumann functional calculus, we get h M i ( ∆ ) h, h i = D Z ψ − i ( ∆ ) ζ i ( x ) M (d x ) h, h E = Z ψ − i ( ∆ ) ζ i ( x ) h M (d x ) h, h i (3.3.10) = Z ψ − i ( ∆ ) ζ i ( x ) D R ∗ E S (d x ) Rh, h E = D R ∗ Z R + χ ψ − i ( ∆ ) ( x ) ζ i ( x ) E S (d x ) Rh, h E = D R ∗ Z R + ζ i ( x ) E S (d x ) E S ( ψ − i ( ∆ )) Z R + ζ i ( x ) E S (d x ) Rh, h E (3.3.9) = D R ∗ ( I + S + . . . + S i − ) E S i ( ∆ )( I + S + . . . + S i − ) Rh, h E (3.3.7) = h R ∗ i E S i ( ∆ ) R i h, h i , h ∈ H , ∆ ∈ B ( R + ) . (3.3.12)Since the operator I + S + . . . + S i − commutes with E S and is invertible in B ( K ),we obtain _ (cid:8) E S i ( ∆ ) R ( R i ) : ∆ ∈ B ( R + ) (cid:9) (3.3.7)&(3.3.9) = _ (cid:8) ( I + S + . . . + S i − ) E S ( ψ − i ( ∆ )) R ( R ) : ∆ ∈ B ( R + ) (cid:9) = ( I + S + . . . + S i − ) _ (cid:8) E S ( ψ − i ( ∆ )) R ( R ) : ∆ ∈ B ( R + ) (cid:9) = ( I + S + . . . + S i − ) _ (cid:8) E S ( ∆ ) R ( R ) : ∆ ∈ B ( R + ) (cid:9) (3.3.11) = K . Hence, by Lemma 3.2.3(a), W { ( S i ) n R ( R i ) : n ∈ Z + } = K . Using (ii) and (3.3.12)and applying the Stone-von Neumann functional calculus to the operator S i , weverify that the equalities (3.2.11) and (3.2.12) hold with ( T i , R i , S i ) in place of( T, R, S ). This shows (iv) and completes the proof. (cid:3)
The following corollary extends the formula (3.2.20) of Corollary 3.2.7 to thecase of powers of conditionally positive definite operators.
Corollary . Suppose T ∈ B ( H ) is a conditionally positive definite op-erator and M is as in Theorem . Then B m ( T i ) = Z R + (1 − x i ) m − (1 + x + . . . + x i − ) M (d x ) , m ≥ , i > . (3.3.13) Proof.
In view of Corollary 3.2.7, it suffices to consider the case i >
2. By theassertions (i) and (ii) of Proposition 3.3.2, T i is conditionally positive definite andthe semispectral measure M i corresponding to T i via Theorem 3.2.5(ii) is given by(3.3.5) and (3.3.6). Using (1.2.4) and the measure transport theorem, we obtain for any Borel function f : R + → C , f ∈ L ( M i ) ⇐⇒ f ◦ ψ i ∈ L ( ˜ M i ) , Z R + f d M i = Z R + f ( x i )(1 + x + . . . + x i − ) M (d x ) , f ∈ L ( M i ) . (3.3.14)Applying (3.2.20) to T i and (3.3.14) to f ( x ) = (1 − x ) m − , we get (3.3.13). (cid:3) As the second application of Theorem 3.3.1, we give a characterization of con-ditionally positive definite operators of class Q (a class of operators having uppertriangular 2 × ]. We also describe the semispectral and the dilation representations ONDITIONALLY POSITIVE DEFINITENESS IN OPERATOR THEORY 33 for such operators. According to Corollary 3.2.7(i), B k ( T ) > k ∈ Z + whenever T is conditionally positive definite. We will show in Proposition 3.3.5below that the single inequality B k ( T ) > k > Q . Following [ ], we say that T ∈ B ( H ) is of class Q if it has a block matrix form T = (cid:20) V E Q (cid:21) with respect to an orthogonal decomposition H = H ⊕ H , where H and H arenonzero Hilbert spaces and V ∈ B ( H ) , E ∈ B ( H , H ) and Q ∈ B ( H ) satisfy V ∗ V = I, V ∗ E = 0 , QE ∗ E = E ∗ EQ and QQ ∗ Q = Q ∗ QQ. (3.3.15)(In particular, by the square root theorem | Q | and | E | commute.) If this is thecase, we write T = (cid:2) V E Q (cid:3) ∈ Q H , H . The Taylor spectrum of a pair ( T , T ) ofcommuting operators T , T ∈ B ( H ) is denoted by σ ( T , T ). It is worth pointingout that in view of [ , Theorem 3.3] for any nonempty compact subset Γ of R and any separable infinite dimensional Hilbert space H , there exist a nonzeroHilbert space H and T = (cid:2) V E Q (cid:3) ∈ Q H , H (relative to H = H ⊕ H ) such that σ ( | Q | , | E | ) = Γ . This important fact enables us to find the spectral region forconditional positive definiteness of operators of class Q (see Proposition 3.3.5 andFigure 1). For a more thorough discussion of these topics the reader is referredto [ ]. Before stating Proposition 3.3.5, we prove an auxiliary lemma which is ofsome independent interest. Lemma . Let
A, B ∈ B ( H ) be two commuting normal operators. Then N ( AB ) = N ( A ) + N ( B ) , (3.3.16) R ( AB ) = R ( A ) ∩ R ( B ) . (3.3.17) Proof.
Let G : B ( C ) → B ( H ) be the joint spectral measure of ( A, B ) (see[ , Theorem 5.21]). Since G ( { (0 , } ) G ( C × { } ) and thus R ( G ( { (0 , } )) ⊆ R ( G ( C × { } )), we obtain R ( G ( C × { } )) = R ( G ( { (0 , } )) + R ( G ( C × { } )) . (3.3.18)Applying the Stone-von Neumann functional calculus and (3.3.18) yields N ( AB ) = N (cid:16) Z C z z d G ( z , z ) (cid:17) = N ( G ( { ( z , z ) ∈ C : z z = 0 } ))= R ( G ( { ( z , z ) ∈ C : z z = 0 } ))= R ( G ( { } × C ∗ )) + R ( G ( C × { } ))= R ( G ( { } × C ∗ )) + R ( G ( { (0 , } )) + R ( G ( C × { } ))= R ( G ( { } × C )) + R ( G ( C × { } )) , (3.3.19)where C ∗ := C \ { } . Similarly, N ( A ) = N (cid:16) Z C z d G ( z , z ) (cid:17) = R ( G ( { } × C )) , (3.3.20) N ( B ) = N (cid:16) Z C z d G ( z , z ) (cid:17) = R ( G ( C × { } )) . (3.3.21) Combining (3.3.19) with (3.3.20) and (3.3.21), we get (3.3.16). Finally, applying(3.3.16) to the adjoints of A and B and taking orthocomplements gives (3.3.17). (cid:3) Proposition . Suppose that T = (cid:2) V E Q (cid:3) ∈ Q H , H . Then the followingconditions are equivalent :(i) T is conditionally positive definite, (ii) σ ( | Q | , | E | ) ⊆ { ( s, t ) ∈ R : s + t } ∪ ([1 , ∞ ) × R + ) , (iii) B k ( T ) > for every ( equivalently, for some ) k ∈ N .Moreover, if T is conditionally positive definite, then the following assertions hold :(a) A := ( I − | Q | − | E | )( I − | Q | ) ∈ B ( H ) + , A commutes with | Q | and M ( ∆ ) = 0 ⊕ √ A P | Q | ( ∆ ) √ A, ∆ ∈ B ( R + ) , (3.3.22) where M is as in Theorem and P | Q | is the spectral measure of | Q | , (b) the representing triplet ( B, C, F ) of T is described by Theorem , (c) the triplet ( K , R, S ) defined below corresponds to T via Theorem K := R ( A ) = R ( I − | Q | − | E | ) ∩ R ( I − | Q | ) , (3.3.23) R ( h ⊕ h ) := √ Ah , h ∈ H , h ∈ H , (3.3.24) S := (cid:0) | Q | (cid:12)(cid:12) K (cid:1) ( K reduces | Q | ) . Proof. (i) ⇔ (ii) Using [ , Proposition 3.10], one can check that T ∗ n B ( T ) T n = (cid:20) Q ∗ n AQ n (cid:21) , n ∈ Z + . (3.3.25)By the square root theorem and (3.3.15), the operators Q , | Q | and | E | commute.Combined with [ , (19)], this implies that A = A ∗ and Q ∗ n AQ n = Q ∗ n Q n A = | Q | n A = Z R τ n d G, n ∈ Z + , (3.3.26)where G is the joint spectral measure of ( | Q | , | E | ) and τ n : R → R is given by τ n ( s, t ) = (1 − s − t )(1 − s ) s n , s, t ∈ R , n ∈ Z + . (3.3.27)It follows from Proposition 3.1.4, [ , Theorem 2], (3.3.25) and (3.3.26) that T isconditionally positive definite if and only if { R τ n ( s, t ) h G (d s, d t ) h, h i} ∞ n =0 is a Stielt-jes moment sequence for every h ∈ H . By [ , Theorem 2.1(i) & Lemma 4.10],the latter holds if and only if σ ( | Q | , | E | ) ⊆ Ξ , where Ξ := (cid:8) ( s, t ) ∈ R : { τ n ( s, t ) } ∞ n =0 is a Stieltjes moment sequence (cid:9) . In view of (3.3.27), it is easily seen that Ξ = { ( s, t ) ∈ R : (1 − s − t )(1 − s ) > } = { ( s, t ) ∈ R : s + t } ∪ ([1 , ∞ ) × R + ) , which shows that (i) and (ii) are equivalent.(ii) ⇔ (iii) This equivalence is a direct consequence of [ , Theorem 9.2(i)].We now prove the “moreover” part. Assume that (i) holds.(a) Applying the spectral mapping theorem (see e.g., [ , Theorem 2.1]), we get σ ( A ) (3.3.27) = σ ( τ ( | Q | , | E | )) = τ ( σ ( | Q | , | E | )) ⊆ R + , ONDITIONALLY POSITIVE DEFINITENESS IN OPERATOR THEORY 35 which together with A = A ∗ implies that A ∈ B ( H ) + . Now, it is clear that theset function M : B ( R + ) → B ( H ) defined by (3.3.22) is a semispectral measurewith compact support. Since | E | commutes with | Q | , so does A . Using this factand applying the Stone-von Neumann functional calculus, we deduce from (3.3.25),(3.3.26) and the square root theorem that h T ∗ n B ( T ) T n h, h i = h Q ∗ n AQ n h , h i = k ( | Q | ) n/ √ Ah k = Z R + x n h√ AP | Q | (d x ) √ Ah , h i = Z R + x n h M (d x ) h, h i , (1.2.4) = D Z R + x n M (d x ) h, h E , h = h ⊕ h ∈ H , n ∈ Z + . This shows that the condition (iii) of Theorem 3.3.1 holds. Applying the “moreover”part of this theorem completes the proof of (a).(b) Obvious.(c) First, note that by (3.3.24), R ( R ) = R ( √ A ) = R ( A ) = K , (3.3.28)so R is well defined. Since A commutes with | Q | , we see that K reduces | Q | , andthus S = (cid:0) | Q | (cid:12)(cid:12) K (cid:1) ∈ B ( K ) + . Moreover, K reduces P | Q | to the spectral measure P S of S . Using (3.3.22), (3.3.24) and (3.3.28), we easily obtain M ( ∆ ) = R ∗ P S ( ∆ ) R, ∆ ∈ B ( R + ) , K = _ (cid:8) P S ( ∆ ) R ( R ) : ∆ ∈ B ( R + ) (cid:9) . Therefore, in view of the proof of the implication (ii) ⇒ (iv) of Theorem 3.2.5, thetriplet ( K , R, S ) satisfies the condition (iv) of Theorem 3.2.5.It remains to prove the second equality in (3.3.23). To do this, first observethat N ( I − | Q | ) = N ( I − | Q | ), which implies that R ( I − | Q | ) = R ( I − | Q | ).From this and Lemma 3.3.4 it follows easily that the second equality in (3.3.23)holds. This completes the proof. (cid:3) Corollary . Suppose that T = (cid:2) V E Q (cid:3) ∈ Q H , H is conditionally positivedefinite and S is as in Proposition . Then the following conditions areequivalent :(i) S = | Q | , (ii) 1 / ∈ σ p ( | Q | + | E | ) and / ∈ σ p ( | Q | ) . Regarding Proposition 3.3.5, it is worth mentioning that in view of [ , Theo-rem 1.2] the operator T = (cid:2) V E Q (cid:3) ∈ Q H , H is subnormal if and only if σ ( | Q | , | E | ) ⊆ { ( s, t ) ∈ R : s + t } ∪ ([1 , ∞ ) × { } ) . For the reader’s convenience, the spectral regions for subnormality and conditionalpositive definiteness of operators of class Q are illustrated in Figure 1. (0, 0) (1, 0)(0, 1) (0, 0) (1, 0)(0, 1) Figure 1.
Spectral regions for subnormality (left) and conditionalpositive definiteness (right) of operators of class Q . In view of Theorem 1.1.1, any subnormal operator T ∈ B ( H ) has the property that the sequence {k T n h k } ∞ n =0 is positive definite for every h ∈ H . As a consequence, any subnormal operator is conditionally positive definite.The converse implication is not true in general (see [ , Example 5.4]). In fact, anystrict 2-isometry is conditionally positive definite but not subnormal (use Proposi-tion 4.3.1, (1.2.2), [ , Lemma 1.21] and [ , Lemma 1]). In this subsection, we dealwith the problem of finding necessary and sufficient conditions for subnormalitywritten in terms of conditional positive definiteness. Theorem 4.1 in [ ], which isthe first result in this direction formulated for d -tuples of operators, shows that acontraction is subnormal if and only if it is conditionally positive definite. The mainresult of this subsection, namely Theorem 3.4.4, generalizes [ , Theorem 4.1]. Inparticular, it covers the case of strongly stable operators (see Corollary 3.4.5).Our first goal is to characterize those conditionally positive definite operatorsthat are subnormal in terms of the parameters B, C, F appearing in the statement(ii) of Theorem 3.1.1.
Theorem . Let T ∈ B ( H ) be a conditionally positive definite operator and ( B, C, F ) be its representing triplet. Then the following statements are equivalent :(i) T is subnormal, (ii) the triplet ( B, C, F ) satisfies the following conditions :(ii-a) x − ∈ L ( F ) and R R + x − F (d x ) I , (ii-b) x − ∈ L ( F ) and B = R R + x − F (d x ) , (ii-c) C = 0 .Moreover, if (i) holds and G is the semispectral measure of T (see (1.2.5)) , then F = M, where M is as in Theorem , B = Z R + ( x − G ◦ φ − (d x ) ,F ( ∆ ) = Z ∆ ( x − G ◦ φ − (d x ) , ∆ ∈ B ( R + ) , (3.4.1) G ◦ φ − ( ∆ ) = Z ∆ x − F (d x ) + δ ( ∆ ) (cid:16) I − Z R + x − F (d x ) (cid:17) , ∆ ∈ B ( R + ) . ONDITIONALLY POSITIVE DEFINITENESS IN OPERATOR THEORY 37
Proof. (i) ⇒ (ii) It follows from Theorem 1.1.1 that {k T n h k } ∞ n =0 is a Stieltjesmoment sequence for every h ∈ H . Hence, by Theorem 2.2.12, we have Z R + x − h F (d x ) h, h i k h k , h ∈ H , (3.4.2) h Bh, h i = Z R + x − h F (d x ) h, h i , h ∈ H , (3.4.3) h Ch, h i = 0 , h ∈ H . (3.4.4)It follows from (1.2.4), (3.4.2) and (3.4.4) that the conditions (ii-a) and (ii-c) aresatisfied. In turn, (3.4.2) yields x − ∈ L ( F ). Combined with (3.4.3), this im-plies (ii-b).(ii) ⇒ (i) Applying (1.2.4) and Theorem 2.2.12 again, we deduce that the se-quence {k T n h k } ∞ n =0 is positive definite for all h ∈ H . Hence, by Theorem 1.1.1, T is subnormal.The “moreover” part can be deduced straightforwardly from (1.2.5) and thecorresponding part of Theorem 2.2.12 (that F = M follows from (ii-c) and Theo-rem 3.2.5(b)). This completes the proof. (cid:3) Corollary . Let T ∈ B ( H ) be a subnormal operator, G be the semis-pectral measure of T , N be the minimal normal extension of T and F be as inTheorem . Then (i) r ( T ) = k T k = sup (cid:8) | z | : z ∈ supp( G ) (cid:9) , (ii) σ ( N ) = supp( G ) and σ ( N ∗ N ) = {| z | : z ∈ supp( G ) } , (iii) if G ( T ) = 0 , where T = { z ∈ C : | z | = 1 } , then (iii-a) the measures F and G ◦ φ − are mutually absolutely continuous, (iii-b) σ ( N ∗ N ) = supp( F ) , (iii-c) k T k = sup supp( F ) . Proof.
The first equality in (i) is a consequence of (1.2.2). It follows from[ , Proposition 4] that σ ( N ) = supp( G ) , (3.4.5)which gives the first equality in (ii). Using [ , Corollary II.2.17], we obtain k T k = k N k (1.2.2) = r ( N ) (3.4.5) = sup {| z | : z ∈ supp( G ) } . This yields the second equality in (i). The second equality in (ii) follows from(3.4.5) and [ , eq. (14), p. 158]. It remains to prove (iii). According to (3.4.1), F is absolutely continuous with respect to G ◦ φ − . In turn, if ∆ ∈ B ( R + ) is such that F ( ∆ ) = 0, then (3.4.1) implies that G ◦ φ − ( ∆ \ { } ) = 0. Since by assumption G ◦ φ − ( { } ) = 0, we see that G ◦ φ − ( ∆ ) = 0. This means that the measures F and G ◦ φ − are mutually absolutely continuous, therefore (iii-a) holds. As aconsequence, supp( F ) = supp( G ◦ φ − ). Combined with [ , Lemma 3(5)], thisimplies (iii-b). Finally, (iii-c) is a direct consequence of (i), (ii) and (iii-b). (cid:3) Corollary . Let T ∈ B ( H ) be a subnormal operator and M be as inTheorem . Then M = 0 if and only if T is an isometry. Proof. If M = 0, then by Theorem 3.4.1, B = C = 0, so by (3.1.2), T is an isometry. Conversely, if T is an isometry, an application of the identity p h T i = p (1) I , p ∈ C [ X ], gives (3.2.10) with M = 0. (cid:3) Theorem 3.4.4 below gives new necessary and sufficient conditions for subnor-mality. The condition (v) of this theorem comprises the case D = 0 which is notcovered by Proposition 3.1.7. Theorem . Let T ∈ B ( H ) . Then the following conditions are equivalent :(i) T is a subnormal contraction, (ii) T is a conditionally positive definite contraction, (iii) T is conditionally positive definite and the telescopic series ∞ X n =0 (cid:0) k T n +1 h k − k T n h k (cid:1) is convergent in R for every h ∈ H , (iv) T is conditionally positive definite and lim n →∞ ( k T n +1 h k − k T n h k ) = 0 , h ∈ H , (3.4.6)(v) the condition (ii) of Theorem holds with C = 0 , D = 0 , F ([1 , ∞ )) = 0 and − x ) ∈ L ( F ) , where D := B + R [0 ,
1) 11 − x F (d x ) ( or equivalently ifall of this holds with “ − x ∈ L ( F ) ” in place of “ − x ) ∈ L ( F ) ” ) , (vi) the condition (ii) of Theorem holds with D = 0 . Proof.
The implications (i) ⇒ (ii) ⇒ (iii) ⇒ (iv) are obvious because if T is acontraction, then the sequence {k T n h k } ∞ n =0 , being monotonically decreasing, isconvergent in R + for all h ∈ H .(iv) ⇒ (i) This implication can be deduced from Corollary 2.3.4 (applied to γ T,h )and Theorems 3.1.1 and 1.1.1.(iv) ⇒ (v) It follows from Theorem 3.1.5 that (iv) implies the variant of (v) with“ − x ∈ L ( F )”. That − x ) ∈ L ( F ) is a consequence of Theorem 3.4.1 and thefact that (iv) implies (i).(v) ⇒ (vi) Assume that the variant of (v) with “ − x ) ∈ L ( F )” holds. Then,by the Cauchy-Schwarz inequality, − x ∈ L ( F ). Observe that (cf. (2.3.8)) T ∗ n T n (3.1.2) = I + n B + Z [0 , Q n ( x ) n F (d x ) ! (2.2.2) = I − Z [0 , − x n ( x − F (d x ) , n ∈ N . This implies that the pair (
D, F ) with D = 0 satisfies the condition (ii) of Theo-rem 3.1.5.(vi) ⇒ (iv) One can apply Theorem 3.1.5. (cid:3) There are other ways to prove some implications of Theorem 3.4.4. Namely,one can show the implication (iv) ⇒ (ii) by using the “moreover” part of Proposi-tion 3.1.4. In turn, the implication (ii) ⇒ (i) can be deduced from [ , Theorem 3.1]and Corollary 3.2.7(ii). The implication (ii) ⇒ (i) (with a different proof) is a partof the conclusion of [ , Theorem 4.1]. By Theorem 3.4.4, an operator T ∈ B ( H ) issubnormal if and only if there exists α ∈ C \ { } such that the operator αT satisfiesany of the equivalent conditions (ii)-(vi) of Theorem 3.4.4. Corollary . Let T ∈ B ( H ) obey any of the following conditions :(i) the sequence {k T n h k } ∞ n =0 is convergent in R + for every h ∈ H , ONDITIONALLY POSITIVE DEFINITENESS IN OPERATOR THEORY 39 (ii) T is strongly stable, i.e., lim n →∞ k T n h k = 0 for every h ∈ H ([
42, 43 ]) , (iii) r ( T ) < .Then T is conditionally positive definite if and only if T is subnormal. Corollary . Let T ∈ B ( H ) . Then the following are equivalent :(i) T is subnormal, (ii) αT is conditionally positive definite for all α ∈ C , (iii) zero is an accumulation point of the set of all α ∈ C \ { } for which αT is conditionally positive definite, (iv) there exists α ∈ C \ { } such that | α | r ( T ) < and αT is conditionallypositive definite. Corollary . Suppose T ∈ B ( H ) is a non-subnormal conditionally posi-tive definite operator. Then r ( T ) > and αT is not conditionally positive definitefor any nonzero complex number such that | α | r ( T ) < . Regarding Corollary 3.4.7, we refer the reader to Example 4.3.6 for an exampleof a non-subnormal conditionally positive definite operator with r ( T ) = 1. Belowwe apply the above to certain translations of quasinilpotent operators (cf. [ ]). Corollary . Let N ∈ B ( H ) and α ∈ C be such that r ( N ) = 0 and | α | < . Then αI + N is conditionally positive definite if and only if N = 0 . Proof. If αI + N is conditionally positive definite, then, since r ( αI + N ) = | α | <
1, we infer from Corollary 3.4.5 and (1.2.2) that k N k = r ( N ) = 0, whichshows that N = 0. (cid:3) Concerning Corollary 3.4.8, note that if N is a nilpotent operator with index ofnilpotency 2 and α ∈ C is such that | α | = 1, then by [ , Theorem 2.2], αI + N is astrict 3-isometry, so by Proposition 4.3.1, αI + N is conditionally positive definite.It is an open question as to whether there exists a quasinilpotent operator N whichis not nilpotent and such that I + N is conditionally positive definite.According to the above discussion, the class of conditionally positive definiteoperators is not scalable, i.e., it is not closed under the operation of multiplyingby nonzero complex scalars. Among non-scalable classes of operators are thosewhich consist of m -isometric and 2-hyperexpansive operators (see [ , Lemma 1.21]and [ , Lemma 1], respectively). On the other hand, the classes of normaloid,hyponormal and subnormal operators are scalable (see [ ] for more examples).The condition (3.4.6) of Theorem 3.4.4 gives rise to a link between the con-ditional positive definiteness of a (bounded) operator T and the subnormality of(in general unbounded) unilateral weighted shift operators W T,h , h ∈ H , definedbelow. Given an operator T ∈ B ( H ) and a vector h ∈ H , we denote by W T,h the unilateral weighted shift in ℓ with weights { e ( k T n +1 h k −k T n h k ) } ∞ n =0 , that is W T,h = U D
T,h , where U ∈ B ( ℓ ) is the unilateral shift and D T,h is the diagonal(normal) operator in ℓ with the diagonal { e ( k T n +1 h k −k T n h k ) } ∞ n =0 (with respectto the standard orthonormal basis of ℓ ). Then for every h ∈ H , W T,h ∈ B ( ℓ ) if and only if sup n ∈ Z + ( k T n +1 h k − k T n h k ) < ∞ ; if this is the case, then k W T,h k = e sup n ∈ Z + ( k T n +1 h k −k T n h k ) . (3.4.7)In view of (3.4.7), the weighted shift W T,h is bounded for all h ∈ H if and only if T satisfies the condition (3.1.17) of Theorem 3.1.5. As discussed in Remark 4.3.3, there are conditionally positive definite operators T for which W T,h is unboundedfor all nonzero vectors h ∈ H and r ( T ) = 1. We show below that subnormalcontractions T are precisely those for which all weighted shifts W T,h , h ∈ H , arebounded, subnormal and of norm one. For the definition and basic facts aboutunbounded subnormal operators we refer the reader to [
67, 68, 69 ]. Proposition . Let T ∈ B ( H ) . Then the following assertions hold :(i) T is conditionally positive definite if and only if W T,h is subnormal for all h ∈ H , (ii) the following conditions are equivalent :(ii-a) T is a subnormal contraction, (ii-b) T is a conditionally positive definite contraction, (ii-c) W T,h is subnormal and k W T,h k = 1 for all h ∈ H , (ii-d) W T,h is conditionally positive definite and k W T,h k = 1 for all h ∈ H . Proof. (i) By using Lemma 2.1.1 and considering √ th instead of h , we deducethat T is conditionally positive definite if and only if the sequence { e k T n h k } ∞ n =0 ispositive definite for all h ∈ H . Replacing h by T h and using Theorem 2.1.3, wesee that the latter holds if and only if { e k T n h k } ∞ n =0 is a Stieltjes moment sequencefor all h ∈ H . Finally, applying [ , Theorem 4] (with [ , Remark 3.1.4]), weconclude that T is conditionally positive definite if and only if W T,h is subnormalfor all h ∈ H .(ii) The equivalences (ii-a) ⇔ (ii-b) and (ii-c) ⇔ (ii-d) follow from the equivalence(i) ⇔ (ii) of Theorem 3.4.4. Noting first that the sequence {k T n h k } ∞ n =0 is convergentin R + for all h ∈ H whenever T is a contraction and then using (i) and (3.4.7), weget the equivalence (ii-b) ⇔ (ii-c). This completes the proof. (cid:3) We now recapitulate our considerations in Table 1, where D stands for the limitof { T ∗ ( n +1) T n +1 − T ∗ n T n } ∞ n =0 in WOT and “YES/NO”means that both possibil-ities may happen. To get row ➀ apply the Gelfand’s formula for spectral radiusand (3.4.7); row ➁ follows from (1.2.2), (3.4.7) and Proposition 3.1.6; row ➂ is aconsequence of (3.4.7) and Propositions 3.1.4 and 3.4.9 (row ➂ also follows from(3.4.7) and Theorems 3.1.5 and 3.4.4). T is conditionally positive definiteand satisfies ➀ , ➁ or ➂ ? = ⇒ T subnormal r ( T ) ➀ ∃ h : W T,h is not bounded YES/NO > ➁ ∀ h : W T,h is bounded and D = 0 NO ➂ ∀ h : W T,h is bounded and D = 0 YES Table 1.
We close this subsection with a new characterization of completely hyperexpan-sive operators. It can be deduced from [ , Theorem 2] and Lemma 2.1.1 by arguingas in the proof of Proposition 3.4.9(i). Despite the formal similarity, the character-izations given in Propositions 3.4.9 and 3.4.10 are radically different, because allunilateral weighted shifts appearing in Proposition 3.4.10 are contractive. Proposition . An operator T ∈ B ( H ) is completely hyperexpansive ifand only if the unilateral weighted shift on ℓ with weights { e ( k T n h k −k T n +1 h k ) } ∞ n =0 is subnormal for all h ∈ H . ONDITIONALLY POSITIVE DEFINITENESS IN OPERATOR THEORY 41
4. A functional calculus and related matters4.1. A functional calculus.
We begin by discussing the space L ∞ ( M ). Sup-pose M : B ( R + ) → B ( H ) is a semispectral measure. We denote by L ∞ ( M ) theBanach space of all equivalence classes of M -essentially bounded complex Borelfunctions on R + equipped with the M -essential supremum norm (see [ , Appen-dix]; see also [ , Section 12.20]). We customarily regard elements of L ∞ ( M )as functions that are identified by equality a.e. [ M ], the latter meaning “almosteverywhere with respect to M ”. In particular, the norm on L ∞ ( M ) takes the form k f k L ∞ ( M ) = min { α ∈ R + : M ( { x ∈ R + : | f ( x ) | > α } ) = 0) } , f ∈ L ∞ ( M ) . The relationship between L ∞ ( M ) and the classical L ∞ ( µ ) is explained below. If µ is a Borel measure on R + , then L ∞ ( M ) = L ∞ ( µ ) if and only if M and µ are mutually absolutely continuous; if this is the case, then k f k L ∞ ( M ) = k f k L ∞ ( µ ) for every f ∈ L ∞ ( M ) . (4.1.1)As shown in Example 4.1.1 below, it may not be possible to find a Borel probabilitymeasure on R + with respect to which a given semispectral measure is absolutelycontinuous. Example . Let Ω be any uncountable bounded subset of R + and let E : B ( R + ) → B ( H ) be the spectral measure given by E ( ∆ ) = M x ∈ Ω χ ∆ ( x ) I H x , ∆ ∈ B ( R + ) , where each H x is a nonzero Hilbert space. Clearly, the following holds.If ∆ ∈ B ( R + ), then E ( ∆ ) = 0 if and only if ∆ ∩ Ω = ∅ . (4.1.2)Suppose to the contrary that E is absolutely continuous with respect to a finiteBorel measure µ on R + . Then by (4.1.2), µ ( { x } ) > x ∈ Ω , which isimpossible because µ is finite and Ω is uncountable (see [ , Problem 12, p. 12]).Plainly, E is compactly supported and supp( E ) = ¯ Ω . ♦ The situation described in Example 4.1.1 cannot happen when H is separable.What is more, the following statement holds. Suppose H is separable and M : B ( R + ) → B ( H ) is a nonzero semis-pectral measure. Then there exists a Borel probability measure µ on R + such that M and µ are mutually absolutely continuous. (4.1.3)To see this, take an orthonormal basis { e j } j ∈ J of H , where J is a countable indexset. Let { a j } j ∈ J be any system of positive real numbers such that X j ∈ J a j h M ( R + ) e j , e j i = 1 . (4.1.4)(This is possible because M = 0.) Define the Borel measure µ on R + by µ ( ∆ ) = X j ∈ J a j h M ( ∆ ) e j , e j i , ∆ ∈ B ( R + ) . By (4.1.4), µ is a probability measure. If ∆ ∈ B ( R + ) is such that µ ( ∆ ) = 0, then0 = h M ( ∆ ) e j , e j i = k ( M ( ∆ ) / e j ) k , j ∈ J, which implies that M ( ∆ ) = 0. Thus E is absolutely continuous with respect to µ .That µ is absolutely continuous with respect to M is immediate. We now prove the following fact. If M : B ( R + ) → B ( H ) is a nonzero compactly supported semispectralmeasure, then k f k L ∞ ( M ) = k f | Ω k C ( Ω ) for every f ∈ L ∞ ( M ) suchthat f | Ω ∈ C ( Ω ) , where Ω := supp( M ) . (4.1.5)Indeed, the inequality “ ” is obvious. If α ∈ R + is such that M ( { x ∈ R + : | f ( x ) | > α } ) = 0 , then M ( { x ∈ Ω : | f ( x ) | > α } ) = 0 and, because the set { x ∈ Ω : | f ( x ) | > α } isopen in Ω , we deduce that | f ( x ) | α for all x ∈ Ω , which after taking infimumover such α ’s yields the inequality “ > ”. This proves (4.1.5).As a consequence of (4.1.5), we have if f, g ∈ L ∞ ( M ) are such that f | Ω , g | Ω ∈ C ( Ω ) and f = g a.e. [ M ] , then f | Ω = g | Ω . The above discussion shows that (still under the assumptions of (4.1.5)) the mapwhich sends a function g ∈ C ( Ω ) to the equivalence class of any of extensionsof g to a complex Borel function on R + is an isometry from C ( Ω ) to L ∞ ( M ).Therefore, C ( Ω ) can be regarded as a closed vector subspace of L ∞ ( M ); this factplays an important role in Theorem 4.1.2(v) below. As shown in (4.1.3) and (4.1.1),if H is separable and M = 0, then L ∞ ( M ) = L ∞ ( µ ) for some Borel probabilitymeasure on R + , so C ( Ω ) is a separable closed vector subspace of L ∞ ( µ ) (see [ ,Theorem V.6.6]), while, in general, L ∞ ( µ ) is not separable (see [ , Problem 2, p.62]). As is easily seen, the above facts (except for separability of C ( Ω )) are truefor regular Borel semispectral measures on topological Hausdorff spaces.We are now ready to construct an L ∞ ( M )-functional calculus that is built upon the basis of Agler’s hereditary functional calculus. Theorem . Suppose that T ∈ B ( H ) is a conditionally positive definiteoperator. Let M : B ( R + ) → B ( H ) be a compactly supported semispectral measuresatisfying (3.2.10) . Then the map Λ T : L ∞ ( M ) → B ( H ) given by Λ T ( f ) = Z R + f d M, f ∈ L ∞ ( M ) , (4.1.6) is continuous and linear. It has the following properties :(i) Λ T ( q ) = (( X − q ) h T i for every q ∈ C [ X ] , (ii) Λ T is positive , i.e., Λ T ( f ) > whenever f ∈ L ∞ ( M ) and f > a.e. [ M ] , (iii) there exist a Hilbert space K , R ∈ B ( H , K ) and S ∈ B ( K ) + such that (3.2.12) holds and Λ T ( f ) = R ∗ f ( S ) R, f ∈ L ∞ ( M ) , (4.1.7)(iv) k Λ T k = k B ( T ) k , (v) if M is nonzero and Ω := supp( M ) , then C [ X ] is dense in C ( Ω ) in the L ∞ ( M ) -norm, Λ T | C ( Ω ) : C ( Ω ) → B ( H ) is a unique continuous linearmap satisfying (i) , k Λ T | C ( Ω ) k = k Λ T k and (cid:13)(cid:13)(cid:13)(cid:13) n X j =0 α j T ∗ j B ( T ) T j (cid:13)(cid:13)(cid:13)(cid:13) k B ( T ) k sup x ∈ Ω (cid:12)(cid:12)(cid:12)(cid:12) n X j =0 α j x j (cid:12)(cid:12)(cid:12)(cid:12) , { α j } nj =0 ⊆ C , n ∈ Z + . (4.1.8) Since Λ T is a positive map on a commutative C ∗ -algebra L ∞ ( M ), the Stinespring theoremimplies that Λ T is completely positive (see [ , Theorem 4]). ONDITIONALLY POSITIVE DEFINITENESS IN OPERATOR THEORY 43
Proof.
Let ( K , R, E ) be as in Theorem 3.2.4. Since for every ∆ ∈ B ( R + ), E ( ∆ ) = 0 if and only if M ( ∆ ) = 0 (see the proof of [ , Theorem 4.4]), we get L ∞ ( M ) = L ∞ ( E ) and k f k L ∞ ( M ) = k f k L ∞ ( E ) for every f ∈ L ∞ ( M ). (4.1.9)Set S = R R + xE (d x ). Applying (4.1.9) and the Stone-von Neumann functionalcalculus, we deduce from (4.1.6) that (4.1.7) is valid and consequently k Λ T ( f ) k k R k k f ( S ) k = k R k k f k L ∞ ( E ) = k R k k f k L ∞ ( M ) , f ∈ L ∞ ( M ) . Hence, Λ T is a continuous positive linear map such that k Λ T k k R k . (4.1.10)Applying (3.2.10) to p = ( X − q , we deduce that Λ T satisfies (i). Substituting q = f = X into (i) and (4.1.7), we infer from (3.2.2) that B ( T ) = Λ T ( X ) = R ∗ R, (4.1.11)which together with (4.1.10) yields k Λ T k = k R k = k B ( T ) k . Thus, in view ofLemma 3.2.3(a), (i)-(iv) hold.It remains to prove (v). Assume that M is nonzero. It follows from (4.1.5)and the Stone-Weierstrass theorem (or the classical Weierstrass theorem combinedwith Tietze extension theorem) that C [ X ] is dense in C ( Ω ) in the L ∞ ( M )-normand so Λ T | C ( Ω ) : C ( Ω ) → B ( H ) is a unique continuous linear map satisfying thecondition (i). Since by (3.2.2) and (3.2.4), n X j =0 α j T ∗ j B ( T ) T j = (( X − q ) h T i (i) = Λ T ( q ) , (4.1.12)where q = P nj =0 α j X j , we can easily deduce (4.1.8) from (iv) and (4.1.5). Using(4.1.11) and (iv) again, we conclude that k Λ T | C ( Ω ) k = k Λ T k . This proves (v) andthus completes the proof. (cid:3) Before stating a corollary to Theorem 4.1.2, we recall that a monic polynomial p ∈ C [ X ] of degree at least one takes the form (see [ , p. 252]) p = ( X − z ) · · · ( X − z n ) , (4.1.13)where z , . . . , z n ∈ C . What is more, p can be written as p = n X j =0 ( − n − j s n − j ( z , . . . , z n ) X j , (4.1.14)where s = 1 and s , . . . , s n are the elementary symmetric functions in complexvariables z , . . . , z n given by s j ( z , . . . , z n ) = X i <...
Applying (4.1.8) to the polynomial (4.1.14) and using (4.1.13), we get(4.1.15). The estimate (4.1.16) is a direct consequence of (4.1.15). Finally, theestimate (4.1.17) follows from (4.1.16) applied to z = 1 and the identity B n +2 ( T ) = n X j =0 (cid:18) nj (cid:19) ( − j T ∗ j B ( T ) T j , n ∈ Z + . which can be proved straightforwardly by using Agler’s hereditary functional cal-culus (see (3.2.2) and (4.1.12)). The estimate (4.1.17) can also be inferred from theidentity (3.2.20) and the statements (viii) and (ix) of [ , Theorem A.1]. (cid:3) In the case of conditionally positive definite operators, Lemma 3.2.1 takes thefollowing form for q = ( X − . Proposition . Let T ∈ B ( H ) be a conditionally positive definite operatorand let M be as in Theorem . Then the set I T = { q ∈ C [ X ] : (( X − q ) h T i = 0 } , is the ideal in C [ X ] generated by the polynomial w T ∈ C [ X ] defined by w T = if Ω is infinite , Q u ∈ Ω ( X − u ) if Ω is finite and nonempty ,X if Ω = ∅ , or equivalently if T is a -isometry , where Ω := supp( M ) . Moreover, if Ω = { u , . . . , u n } , where n ∈ N and u , . . . , u n are distinct, then the following identity holds n X j =0 ( − j s n − j ( u , . . . , u n ) T ∗ j B ( T ) T j = 0 . Proof.
First note that by (4.1.6) and Theorem 4.1.2(i), Z Ω p ( x ) M (d x ) = Λ T ( p ) = (( X − p ) h T i , p ∈ C [ X ] . (4.1.18)It follows from Lemma 3.2.1 that I T is an ideal in C [ X ] generated by some polyno-mial w ∈ C [ X ]. Applying (4.1.18) to p = w ∗ w , we see that R Ω | w ( x ) | M (d x ) = 0,and so, by (1.2.4), we have w | Ω = 0 . (4.1.19)We now consider three cases. Case 1
The set Ω is infinite.Since nonzero polynomials may have only finite number of roots, we deducefrom (4.1.19) that w = 0. Case 2
The set Ω is empty (or equivalently, by Proposition 4.3.1, T is a 2-isometry).Then, in view of (4.1.18), I T = C [ X ] and so X generates the ideal I T . Case 3
The set Ω is finite and nonempty. ONDITIONALLY POSITIVE DEFINITENESS IN OPERATOR THEORY 45
Set w T = Q u ∈ Ω ( X − u ). Clearly, by (4.1.18), w T ∈ I T . It follows from thefundamental theorem of algebra (see [ , Theorem V.3.19]) and (4.1.19) that thepolynomial w T divides w . Since w generates the ideal I T , w divides w T and so w T = αw , where α ∈ C \ { } . This means that w T generates I T .The “moreover” part is a direct consequence of (4.1.15). (cid:3) Theorem 4.1.2(i), Proposition 4.1.4 and (4.1.5) lead to the following corollary.
Corollary . Let T ∈ B ( H ) be a conditionally positive definite operatorand let M and Λ T be as in Theorem . Then the following assertions hold :(i) the map C [ X ] ∋ q (( X − q ) h T i ∈ B ( H ) is injective if and only if supp( M ) is infinite, (ii) if q ∈ C [ X ] is such that Λ T ( q ) = 0 , then q = 0 a.e. [ M ] , which means thatthe restriction of Λ T to equivalence classes of polynomials is injective. In Subsection 4.1 we were discussing theaction of the functional calculus established in Theorem 4.1.2 on polynomials. Inthis subsection we concentrate on showing how this functional calculus may workin the case of real analytic functions.Let T , M and Λ T be as in Theorem 4.1.2. Assume that M is nonzero. Set Ω = supp( M ) . Suppose that P ∞ n =0 a n x n is a power series in the real variable x with complex coefficients a n such thatlim sup n →∞ | a n | /n < Ω (cid:16) with 10 = ∞ (cid:17) . (4.2.1)Then the series P ∞ n =0 a n x n is uniformly convergent on [0 , sup Ω ] to a continuousfunction on Ω , say f . Hence by Theorem 4.1.2, we have Λ T ( f ) = ∞ X n =0 a n Λ T ( X n ) (4.1.12) = ∞ X n =0 a n T ∗ n B ( T ) T n . (4.2.2)Let ( K , R, S ) be as Theorem 4.1.2(iii). In particular, (3.2.12) holds and Λ T ( f ) (4.1.7) = R ∗ f ( S ) R. Combined with (4.2.2), this implies that ∞ X n =0 a n T ∗ n B ( T ) T n = R ∗ f ( S ) R. (4.2.3)It follows from Theorem 4.1.2(i) and (4.1.7) that (3.3.2) holds. According to theproof of the implication (ii ′ ) ⇒ (i) of Theorem 3.3.1, (3.2.11) holds, so by Theo-rem 3.2.5(c), σ ( S ) = Ω and k S k = sup Ω . Since the map C ( σ ( S )) ∋ g g ( S ) ∈ B ( H ) is a unital isometric ∗ -homomorphism (see [ , Theorem VIII.2.6]), we get(see also (4.1.5) and (4.1.9)) f ( S ) = ∞ X n =0 a n S n . (4.2.4)Concerning (4.2.2), note that ∞ X n =0 a n T ∗ n B ( T ) T n = ∞ X n =0 a n ∇ nT ( B ( T )) . Since r ( ∇ T ) = r ( T ) (a general fact which follows from Gelfand’s formula forspectral radius), we deduce that the series P ∞ n =0 a n ∇ nT converges in B ( B ( H )) iflim sup n →∞ | a n | /n < r ( T ) . The last inequality is in general stronger than (4.2.1)because by Theorem 3.2.5(c), 1 r ( T ) Ω .
Let us now discuss two important cases. We begin with a n = z n for every n ∈ Z + , where z ∈ C . Then the above considerations lead to ∞ X n =0 z n T ∗ n B ( T ) T n ( † ) = R ∗ ( I − zS ) − R, z ∈ C , | z | < Ω , (4.2.5)where ( † ) follows from (4.2.1), (4.2.3), (4.2.4) and the Carl Neumann theorem (see[ , Theorem 10.7]). In particular, the following estimate holds (see (4.1.11)) (cid:13)(cid:13)(cid:13) ∞ X n =0 z n T ∗ n B ( T ) T n (cid:13)(cid:13)(cid:13) | z − | k B ( T ) k dist( z − , Ω ) , z ∈ C , < | z | < Ω .
In view of the previous paragraph and (4.2.5), we have( I − z ∇ T ) − ( B ( T )) = R ∗ ( I − zS ) − R, z ∈ C , | z | < r ( T ) , where I is the identity map on B ( H ). Note also that if (4.2.5) holds, then bydifferentiating the operator valued functions appearing on both sides of the equality(4.2.5) n times at 0, we obtain (3.3.4), which by Theorem 3.3.1 implies that T isconditionally positive definite.It is a matter of routine to show that for an operator T ∈ B ( H ) the operatorvalued function appearing on the left-hand side of the equality (4.2.5), call it Ψ ,is uniquely determined by the requirement that it be an analytic B ( H )-valuedfunction defined on an open disk D r = { z ∈ C : | z | < r } for some r ∈ (0 , ∞ )such that Ψ ( z ) = B ( T ) + zT ∗ Ψ ( z ) T, z ∈ D r . (4.2.6)In other words, we have proved that T is conditionally positive definite if and onlyif there exists r ∈ (0 , ∞ ) such that the analytic function Ψ associated with T via(4.2.6) satisfies the following equation Ψ ( z ) = R ∗ ( I − zS ) − R, z ∈ D r , for some triplet ( K , R, S ) consisting of a Hilbert space K , an operator R ∈ B ( H , K )and a positive operator S ∈ B ( K ) such that r k S k a n = z n n ! for every n ∈ Z + , where z ∈ C , then ∞ X n =0 z n n ! T ∗ n B ( T ) T n ( ‡ ) = R ∗ e zS R, z ∈ C , where ( ‡ ) is a consequence of (4.2.1), (4.2.3) and (4.2.4), or equivalently thate z ∇ T ( B ( T )) = R ∗ e zS R, z ∈ C . ONDITIONALLY POSITIVE DEFINITENESS IN OPERATOR THEORY 47
In particular, we have ∞ X n =0 i n x n n ! T ∗ n B ( T ) T n = R ∗ e i xS R, x ∈ R . (4.2.7)Since { e i xS } x ∈ R is a uniformly continuous group of unitary operators, we obtain (cid:13)(cid:13)(cid:13)(cid:13) ∞ X n =0 i n x n n ! T ∗ n B ( T ) T n (cid:13)(cid:13)(cid:13)(cid:13) (4.2.7) k R k = k B ( T ) k , x ∈ R , or equivalently k e ix ∇ T ( B ( T )) k k B ( T ) k , x ∈ R . As in the previous case, we observe that if (4.2.7) holds, then by differentiating theoperator valued functions appearing on both sides of the equality (4.2.7) n timesat 0, we obtain (3.3.4), which as we know implies that T is conditionally positivedefinite. In view of Subsection 4.2, the natural question arisesof when the closed support of the semispectral measure M associated with a givenconditionally positive definite operator T via Theorem 3.2.5(ii) is equal to ∅ , { } or { } . Surprisingly, the answers to this seemingly simple question that are givenin Propositions 4.3.1 and 4.3.5 (see also Corollary 3.4.3) lead to three relativelybroad classes of operators, including 2- and 3-isometries. The fact that 3-isometriesare conditionally positive definite was already proved in [ , Proposition 2.7]. InProposition 4.3.1 below, F and M denote the semispectral measures appearing inTheorems 3.1.1(ii) and 3.2.5(ii), respectively. Proposition . Let T ∈ B ( H ) . Then (i) if T is conditionally positive definite, then M = 0 if and only if Λ T = 0 ,or equivalently if and only if T is a -isometry, (ii) the following conditions are equivalent :(a) T is conditionally positive definite and F = 0 , (b) T is conditionally positive definite and supp( M ) ⊆ { } , (c) T ∗ n T n = I − n B ( T ) + n ( n − B ( T ) for all n ∈ Z + , (d) T is a -isometry.Moreover, if an m -isometry is conditionally positive definite, then it is a -isometry. Proof. (i) The first equivalence in (i) follows from (4.1.6) by considering char-acteristic functions, while the second is a direct consequence of Theorem 4.1.2(iv).(ii) The implication (a) ⇒ (c) follows from (3.1.2). Straightforward computa-tions shows that the implication (c) ⇒ (d) holds. If (d) holds, then for all n > T ∗ n T n = (( X −
1) + 1) n h T i = n X j =0 (cid:18) nj (cid:19) ( X − j h T i ( ∗ ) = X j =0 (cid:18) nj (cid:19) ( − j B j ( T ) , where ( ∗ ) follows from Remark 3.2.2. This yields (c). If (c) holds, then the right-hand side of the equality in (c) is nonnegative for all n ∈ Z + , which implies that B ( T ) >
0. Clearly (3.1.2) holds with B = − ( B ( T ) + B ( T )), C = B ( T ) and F = 0, so by Theorem 3.1.1, (a) holds. By (3.2.13), (a) and (b) are equivalent.The “moreover” part follows from [ , Theorem 3.3] and Proposition 2.2.11. (cid:3) As shown below the class of 2-isometries is the intersection of the classes ofconditionally positive definite and 2-hyperexpansive operators.
Proposition . If T ∈ B ( H ) , then the following conditions are equivalent :(i) T is a -isometry, (ii) T is completely hyperexpansive and conditionally positive definite, (iii) T is -hyperexpansive and conditionally positive definite. Proof. (i) ⇒ (ii) By [ , Lemma 1], any 2-hyperexpansive operator T is ex-pansive, i.e., B ( T )
0. This and Remark 3.2.2 implies that any 2-isometry iscompletely hyperexpansive. In view of Proposition 4.3.1, (ii) is valid.(ii) ⇒ (iii) This is obvious.(iii) ⇒ (i) Applying (3.2.20) to m = 2, we see that M ( R + )
0, which impliesthat M = 0, so again by (3.2.20) with m = 2, B ( T ) = 0 showing that T is a2-isometry. This completes the proof. (cid:3) The implication (ii) ⇒ (i) of Proposition 4.3.2 follows also from [ , Theorem 2].The above enables us to make several comments related to Theorem 3.1.5 andProposition 3.1.8. Remark . a) First, note that each 2-isometry T ∈ B ( H ) satisfies thecondition (i) of Theorem 3.1.5. Indeed, by Proposition 4.3.1, T is conditionallypositive definite and T ∗ ( n +1) T n +1 − T ∗ n T n = B, n ∈ Z + , where B = − B ( T ).b) Suppose that T ∈ B ( H ) is a strict 3-isometry. Then, by Proposition 4.3.1, T is conditionally positive definite. However, T does not satisfy the assumption(3.1.17) of Theorem 3.1.5. In fact, we can show more. By (3.2.20) and Proposi-tion 4.3.1, B ( T ) > T ∗ ( n +1) T n +1 − T ∗ n T n = − B ( T ) + n B ( T ) for all n ∈ Z + .This yieldssup n ∈ Z + ( k T n +1 h k − k T n h k ) = ( −h B ( T ) h, h i if h ∈ N ( B ( T )) , ∞ if h ∈ H \ N ( B ( T )) . (4.3.1)Since T is not a 2-isometry, N ( B ( T )) = H , so T does not satisfy (3.1.17).c) It turns out that there are strict 3-isometries T such that N ( B ( T )) = { } .Indeed, let W be the unilateral weighted shift on ℓ with weights (cid:8) √ n +3 √ n +1 (cid:9) ∞ n =0 . Itfollows from [ , Proposition 8] and [ , Lemma 1.21] that W is a strict 3-isometryfor which r ( T ) = 1. We claim that N ( B ( W )) = { } . (4.3.2)Indeed, it is a matter of routine to verify that B ( W ) is the diagonal opera-tor (with respect to the the standard orthonormal basis of ℓ ) with the diagonal (cid:8) n +1)( n +2) (cid:9) ∞ n =0 , which yields (4.3.2). In particular, (4.3.2) implies that W is astrict 3-isometry and, by (4.3.1),sup n ∈ Z + ( k W n +1 h k − k W n h k ) = ∞ , h ∈ ℓ \ { } . ONDITIONALLY POSITIVE DEFINITENESS IN OPERATOR THEORY 49 d) Let W be the unilateral weighted shift as in c). Then W is a 3-isometryand, by Proposition 4.3.1, we have W ∗ n W n = I + nB + n C, n ∈ Z + , (4.3.3)where B = − (cid:0) B ( W ) + B ( W ) (cid:1) and C = B ( W ). We easily check that B and C are diagonal operators with diagonals (cid:8) n +3( n +1)( n +2) (cid:9) ∞ n =0 and (cid:8) n +1)( n +2) (cid:9) ∞ n =0 ,respectively, so B > C > N ( B ) = N ( C ) = { } . By (4.3.3), k W n k α · n for all n ∈ N , where α = p k B k + k C k . We show that there are no ε ∈ (0 , ∞ )and β ∈ R + such that k W n k β · n − ε for all n ∈ N . Indeed, otherwise we have h Ch, h i h ( I + nB + n C ) h, h i n = k W n h k n β k h k n ε , n ∈ N , h ∈ ℓ , which contradicts N ( C ) = { } . ♦ We now turn to the case when supp( M ) = { } . We first prove a result that is ofsome independent interest (see [ , Proposition 8] for the case of weighted shifts). Lemma . Suppose that the restriction of an operator T ∈ B ( H ) to R ( T ) is subnormal. Then T is subnormal if and only if Z R + t d µ h ( t ) for all h ∈ H such that k h k = 1 , (4.3.4) where µ h stands for the ( unique ) representing measure of the Stieltjes moment se-quence {k T n +1 h k } ∞ n =0 . Proof.
Applying Theorem 1.1.1 to T | R ( T ) and using Lemma 2.1.2, we seethat the sequence {k T n +1 h k } ∞ n =0 is a determinate Stieltjes moment sequence forevery h ∈ H . By Theorem 1.1.1, T is subnormal if and only if for every h ∈ H for which k h k = 1, the sequence {k T n h k } ∞ n =0 is a Stieltjes moment sequence, orequivalently, by [ , Lemma 6.1.2], if and only if the condition (4.3.4) holds. (cid:3) Proposition . For T ∈ B ( H ) , the following conditions are equivalent :(i) T is conditionally positive definite and supp( M ) = { } , where M is as inTheorem , (ii) B ( T ) T = 0 , B ( T ) > and B ( T ) = 0 , (iii) T ∗ n T n = I − B ( T ) + n ( B ( T ) − B ( T )) for all n ∈ N , B ( T ) > and B ( T ) = 0 , (iv) T satisfies Theorem with supp( F ) = { } .Moreover, if (i) holds, then (a) r ( T ) = 1 whenever T = 0 , (b) T is subnormal if and only if B ( T ) T = 0 and k T k if this is thecase, then k T k = 1 provided T = 0 . Proof. (i) ⇒ (ii) Substituting q = X into (4.1.12) yields T ∗ B ( T ) T = 0. ByCorollary 3.2.7, B ( T ) = M ( R + ) >
0. Putting this all together implies (ii).(ii) ⇒ (i) Note that the set function M : B ( R + ) → B ( H ) defined by M ( ∆ ) = χ ∆ (0) B ( T ) for ∆ ∈ B ( R + ) is a semispectral measure such that supp( M ) = { } .Clearly (3.3.3) holds, so by Theorem 3.3.1, T is conditionally positive definite and(3.2.10) is valid. (i) ⇒ (iii) Let ( B, C, F ) be the representing triplet of T . According to Theo-rem 3.2.5(b), F = M , C = 0 and B = − B ( T ), so by (3.1.2) and Corollary 3.2.7, T ∗ n T n = I − n B ( T ) + Q n (0) B ( T )= I − n B ( T ) + ( n − B ( T ) , n ∈ N . This together with the implication (i) ⇒ (ii) gives (iii).(iii) ⇒ (iv) As above, the set function F : B ( R + ) → B ( H ) defined by F ( ∆ ) = χ ∆ (0) B ( T ) for ∆ ∈ B ( R + ) is a semispectral measure for which supp( F ) = { } .Set D = B ( T ) − B ( T ). It is easily seen that D and F satisfy Theorem 3.1.5(ii).(iv) ⇒ (i) Apply Theorems 3.1.5 and 3.2.5(b).We now prove the “moreover” part.(a) If D = 0, then by Theorem 3.1.5 and Proposition 3.1.7, r ( T ) = 1. Supposethat D = 0. Then by (iii), T ∗ n T n = I − B ( T ) for all n ∈ N . This together with T = 0 implies that I − B ( T ) = 0, so by Gelfand’s formula for spectral radius r ( T ) = 1.(b) Suppose first that T is subnormal. It follows from (iii) that for every h ∈ H , k T n h k = h ( I − B ( T )) h, h i + n h ( B ( T ) − B ( T )) h, h i , n ∈ N . (4.3.5)By Theorem 1.1.1, {k T n +1 h k } ∞ n =0 is a Stieltjes moment sequence for every h ∈ H .Combined with (4.3.5) and [ , Lemma 4.7], this implies that h ( B ( T ) − B ( T )) h, h i = 0 , h ∈ H , or equivalently that T ∗ B ( T ) T = 0. By (a) and (1.2.2), T is a contraction (in fact, k T k = 1 if T = 0), so B ( T ) > B ( T ) T = 0.In turn, if k T k B ( T ) T = 0, then T is a contraction whose restrictionto R ( T ) is an isometry, so an application of Lemma 4.3.4 with µ h := k T h k δ shows that T is subnormal. This completes the proof. (cid:3) Now we give an example of an operator satisfying the condition (i) of Proposi-tion 4.3.5. In particular, we show that the class of operators satisfying this conditioncan contain both (non-isometric) subnormal and non-subnormal operators.
Example . Fix real numbers a ∈ (0 , ∞ ) and b ∈ [1 , ∞ ) such that θ := 1 − a + ab > . (4.3.6)Define the sequence { λ n } ∞ n =0 ⊆ (0 , ∞ ) by λ n = √ a if n = 0 , q n ( b − n − b − if n > . Let W be the unilateral weighted shift on ℓ with weights { λ n } ∞ n =0 . It follows from[ , Lemma 6.1 & Proposition 6.2(iii)] that W ∈ B ( ℓ ) and k W k = max (cid:8) a, b (cid:9) . (4.3.7)One can also verify that B ( W ) is the diagonal operator (with respect to the thestandard orthonormal basis of ℓ ) with the diagonal ( θ, , , . . . ). This togetherwith (4.3.6) implies that B ( W ) W = 0, B ( W ) > k B ( W ) k = θ > W satisfies the condition (i) of thisproposition. Taking b = a > W on a explicit, we ONDITIONALLY POSITIVE DEFINITENESS IN OPERATOR THEORY 51 see that k W a k = √ a > k B ( W a ) k = ( a − . Since, by Proposition 4.3.5(a), r ( W a ) = 1 for every a ∈ (1 , ∞ ), we deduce that W a is not normaloid for all a > a →∞ k W a k = lim a →∞ k B ( W a ) k = ∞ .In turn, if a ∈ (0 ,
1) and b = 1, then one can verify that B ( W ) W = 0 andby (4.3.7), k W k = 1, so by Proposition 4.3.5, the operator W is subnormal and r ( W ) = 1. ♦ We conclude this subsection with a remark related to Proposition 4.3.5 andExample 4.3.6.
Remark . Suppose that T ∈ B ( H ) is nonzero and satisfies the condition(i) of Proposition 4.3.5 (the zero operator on nonzero H does satisfy (i)). Accordingto the condition (iii) of this proposition, T ∗ n T n is a polynomial in n if n varies over N however not when n varies over Z + . Indeed, otherwise, since a nonzero polynomialmay have only finite number of roots, we deduce from (iii) that I = I − B ( T ),which contradicts B ( T ) = 0. In other words, in view of [ , p. 389] (see also[ , Theorem 3.3]), the requirement that T ∗ n T n be a polynomial in n if n variesover N is not enough for T to be an m -isometry no matter what is m . Finallynote that T falls under Case 3 of the proof of Theorem 3.2.5(c) and the discussionperformed in Remark 3.2.8a). Indeed, by Theorem 3.2.5(b), Proposition 4.3.5(a)and Corollary 3.2.7, we see that B = − B ( T ), C = 0, F = M , ϑ := sup supp( F ) =0, r ( T ) = 1 and D := B + Z R + − x F (d x ) = B ( T ) − B ( T ) . Moreover, in view of Example 4.3.6, both cases D = 0 and D = 0 can appear. ♦ Acknowledgement . A substantial part of this paper was written while thefirst and the third author visited Kyungpook National University during the au-tumns of 2018 and 2019. They wish to thank the faculty and the administration ofthis unit for their warm hospitality.
References [1] J. Agler, Hypercontractions and subnormality,
J. Operator Theory (1985), 203-217.[2] J. Agler, A disconjugacy theorem for Toeplitz operators, Amer. J. Math. (1990), 1-14.[3] J. Agler, M. Stankus, m -isometric transformations of Hilbert spaces, I, Integr. Equ. Oper.Theory (1995), 383-429.[4] J. Agler, M. Stankus, m -isometric transformations of Hilbert spaces, II, Integr. Equ. Oper.Theory (1995), 1-48.[5] J. Agler, M. Stankus, m -isometric transformations of Hilbert spaces, III, Integr. Equ. Oper.Theory (1996), 379-421.[6] R. B. Ash, Probability and measure theory , Harcourt/Academic Press, Burlington, 2000.[7] A. Athavale, Some operator theoretic calculus for positive definite kernels,
Proc. Amer. Math.Soc. (1991), 701-708.[8] A. Athavale, On completely hyperexpansive operators,
Proc. Amer. Math. Soc. (1996),3745-3752.[9] A. Athavale, The complete hyperexpansivity analog of the Embry conditions,
Studia Math. (2003), 233-242.[10] C. Berg, J. P. R. Christensen, P. Ressel,
Harmonic Analysis on Semigroups , Springer-Verlag,Berlin 1984.[11] T. Berm´udez, A. Martin´on, J. A. Noda, An isometry plus a nilpotent operator is an m -isometry. Applications, J. Math. Anal. Appl. (2013), 505-512. [12] M. Sh. Birman, M. Z. Solomjak,
Spectral theory of selfadjoint operators in Hilbert space , D.Reidel Publishing Co., Dordrecht, 1987.[13] T. M. Bisgaard, Positive definite operator sequences,
Proc. Amer. Math. Soc. (1994),1185-1191.[14] T. M. Bisgaard, Z. Sasv´ari,
Characteristic functions and moment sequences. Positive defi-niteness in probability,
Nova Science Publishers, Inc., Huntington, NY, 2000.[15] F. Botelho, J. Jamison, Isometric properties of elementary operators,
Linear Algebra Appl. (2010), 357-365.[16] P. Budzy´nski, Z. J. Jab lo´nski, I. B. Jung, J. Stochel, Unbounded subnormal compositionoperators in L -spaces, J. Funct. Anal. (2015), 2110-2164.[17] P. Budzy´nski, Z. J. Jab lo´nski, I. B. Jung, J. Stochel, Subnormality of unbounded compositionoperators over one-circuit directed graphs: exotic examples,
Adv. Math. (2017), 484-556.[18] P. Budzy´nski, Z. J. Jab lo´nski, I. B. Jung, J. Stochel, Unbounded weighted compositionoperators in L -spaces, Lect. Notes Math. , Volume 2209, Springer 2018.[19] S. Chavan, Z. J. Jab lo´nski, I. B. Jung, J. Stochel, Taylor spectrum approach to Brownian-typeoperators with quasinormal entry,
Ann. Mat. Pur. Appl. , https://doi.org/10.1007/s10231-020-01018-w[20] S. Chavan, V. M. Sholapurkar, Completely monotone functions of finite order and Agler’sconditions,
Studia Math. (2015), 229-258.[21] S. Chavan, V. M. Sholapurkar, Completely hyperexpansive tuples of finite order,
J. Math.Anal. Appl. (2017), 1009-1026.[22] D. Cicho´n, Jan Stochel, Subnormality, analyticity and perturbations,
Rocky Mountain J.Math. (2007), 1831-1869.[23] J. B. Conway, A course in functional analysis , Graduate Texts in Mathematics , Springer-Verlag, New York, 1990.[24] J. B. Conway, The theory of subnormal operators , Mathematical Surveys and Monographs, , American Mathematical Society, Providence, RI, 1991.[25] R. E. Curto, Quadratically hyponormal weighted shifts, Integr. Equ. Oper. Theory (1990),49-66.[26] R. Curto and M. Putinar, Nearly subnormal operators and moment problems, J. Funct. Anal. (1993), 480-497.[27] J. Daneˇs, On local spectral radius, ˇCasopis Pˇest. Mat. (1987), 177-187.[28] M. R. Embry, A generalization of the Halmos-Bram criterion for subnormality,
Acta Sci.Math. ( Szeged ) (1973), 61-64.[29] G. Exner, I. B. Jung, C. Li, On k -hyperexpansive operators, J. Math. Anal. Appl. (2006),569-582.[30] T. Furuta, Invitation to linear operators, Taylor & Francis, Ltd., London, 2001.[31] C. Gu, On ( m, p )-expansive and ( m, p )-contractive operators on Hilbert and Banach spaces,
J. Math. Anal. Appl. (2015), 893-916.[32] P. R. Halmos,
A Hilbert space problem book , Springer-Verlag, New York Inc. 1982.[33] R. A. Horn, C. R. Johnson,
Matrix analysis , Cambridge University Press, Cambridge, 1985.[34] T. W. Hungerford,
Algebra , Graduate Texts in Mathematics 73, Springer-Verlag, New York,1974.[35] Z. Jab lo´nski, Complete hyperexpansivity, subnormality and inverted boundedness conditions,
Integr. Equ. Oper. Theory (2002), 316-336.[36] Z. J. Jab lo´nski, I. B. Jung, J. Stochel, Weighted shifts on directed trees, Memoirs of the AMS (2012).[37] Z. J. Jab lo´nski, I. B. Jung, J. Stochel, A non-hyponormal operator generating Stieltjes mo-ment sequences,
J. Funct. Anal. (2012), 3946-3980.[38] Z. J. Jab lo´nski, I. B. Jung, J. Stochel, m -Isometric operators and their local properties, LinearAlgebra Appl. (2020), 49-70.[39] Z. Jab lo´nski, J. Stochel, Unbounded 2-hyperexpansive operators,
Proc. Edin. Math. Soc. (2001), 613-629.[40] I. Jung, J. Stochel, Subnormal operators whose adjoints have rich point spectrum, J. Funct.Anal. (2008), 1797-1816.[41] A. N. Kolmogoroff, Stationary sequences in Hilbert’s space, (Russian)
Bolletin MoskovskogoGosudarstvenogo Universiteta. Matematika (1941), 40pp. ONDITIONALLY POSITIVE DEFINITENESS IN OPERATOR THEORY 53 [42] C. S. Kubrusly, Strong stability does not imply similarity to a contraction,
Systems ControlLett. (1990), 397-400.[43] C. S. Kubrusly, An introduction to models and decompositions in operator theory , Birkh¨auserBoston, Inc., Boston, MA, 1997.[44] A. Lambert, Subnormality and weighted shifts,
J. London Math. Soc. (1976), 476-480.[45] A. Lambert, Subnormal composition operators, Proc. Am. Math. Soc. , 750-754 (1988).[46] P. Masani, On helixes in Hilbert space. I,
Teor. Verojatnost. i Primenen. (1972), 3-20.[47] S. McCullough and V. I. Paulsen, A note on joint hyponormality, Proc. Amer. Math. Soc. (1989), 187-195.[48] W. Mlak,
Dilations of Hilbert space operators ( general theory ), Dissertationes Math. (1978), 61p.[49] W. Mlak, Conditionally positive definite functions on linear spaces, Ann. Polon. Math. (1983), 187-239.[50] K. R. Parthasarathy, K. Schmidt, Positive definite kernels, continuous tensor products, andcentral limit theorems of probability theory , Lecture Notes in Mathematics, Vol. 272, Springer-Verlag, Berlin-New York, 1972.[51] S. Richter, Invariant subspaces of the Dirichlet shift,
Jour. Reine Angew. Math. (1988),205-220.[52] W. Rudin,
Functional analysis,
McGraw-Hill Series in Higher Mathematics, McGraw-HillBook Co., New York, 1973.[53] W. Rudin,
Principles of mathematical analysis , International Series in Pure and AppliedMathematics, McGraw-Hill Book Co., New York, 1976.[54] W. Rudin,
Real and Complex Analysis , McGraw-Hill Book Co., New York, 1987.[55] Z. Sasv´ari,
Multivariate characteristic and correlation functions,
De Gruyter Studies in Math-ematics, 50, Walter de Gruyter & Co., Berlin, 2013.[56] K. Schm¨udgen,
Unbounded self-adjoint operators on Hilbert space,
Graduate Texts in Math-ematics, 265, Springer, Dordrecht, 2012.[57] I. Schur, Bemerkungen zur Theorie der beschr¨ankten Bilinearformenz mit unendlich vielenVer¨anderlichen,
J. Reine Angew. Math. (1911), 1-29.[58] V. M. Sholapurkar, A. Athavale, Completely and alternatingly hyperexpansive operators,
J.Operator Theory (2000), 43-68.[59] B. Simon, The classical moment problem as a self-adjoint finite difference operator, Adv.Math. (1998), 82-203.[60] T. Stieltjes, Recherches sur les fractions continues,
Anns. Fac. Sci. Univ. Toulouse (1894-1895), J1-J122; , A5-A47.[61] W. F. Stinespring, Positive functions on C ∗ -algebras, Proc. Amer. Math. Soc. (1955),211-216.[62] J. Stochel, The Fubini theorem for semi-spectral integrals and semi-spectral representationsof some families of operators, Univ. Iagel. Acta Math. (1987), 17-27.[63] J. Stochel, Seminormal composition operators on L spaces induced by matrices, HokkaidoMath. J. (1990), 307-324.[64] J. Stochel, Characterizations of subnormal operators, Studia Math. (1991), 227-238.[65] J. Stochel, Decomposition and disintegration of positive definite kernels on convex ∗ -semigroups, Ann. Polon. Math. (1992), 243-294.[66] J. Stochel, J. B. Stochel, Composition operators on Hilbert spaces of entire functions withanalytic symbols, J. Math. Anal. Appl. (2017), 1019-1066.[67] J. Stochel, F. H. Szafraniec, On normal extensions of unbounded operators. I,
J. OperatorTheory (1985), 31-55.[68] J. Stochel, F. H. Szafraniec, On normal extensions of unbounded operators. II, Acta. Sci.Math. ( Szeged ) (1989), 153-177.[69] J. Stochel, F. H. Szafraniec, On normal extensions of unbounded operators. III. Spectralproperties, Publ. RIMS, Kyoto Univ. (1989), 105-139.[70] B. Sz.-Nagy, A moment problem for self-adjoint operators, Acta Math. Acad. Sci. Hungar. (1953), 285-293.[71] A. E. Taylor, D. C. Lay,
Introduction to functional analysis,
Second edition, John Wiley &Sons, New York-Chichester-Brisbane, 1980.
Instytut Matematyki, Uniwersytet Jagiello´nski, ul. Lojasiewicza 6, PL-30348 Kra-k´ow, Poland
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