Resolvent criteria for similarity to a normal operator with spectrum on a curve
aa r X i v : . [ m a t h . F A ] M a r RESOLVENT CRITERIA FOR SIMILARITY TO ANORMAL OPERATOR WITH SPECTRUM ON A CURVE
MICHAEL A. DRITSCHEL, DANIEL EST´EVEZ, AND DMITRY YAKUBOVICH
Abstract.
We give some new criteria for a Hilbert space operator withspectrum on a smooth curve to be similar to a normal operator, interms of pointwise and integral estimates of the resolvent. These resultsgeneralize criteria of Stampfli, Van Casteren and Naboko, and answersseveral questions posed by Stampfli in [48]. The main tools are from ourrecent results [12] on dilation to the boundary of the spectrum, alongwith the Dynkin functional calculus for smooth functions, which is basedon pseudoanalytic continuation. Introduction
As Stampfli proved in 1969 (see [47]), if Γ ⊂ C is a smooth curve, T is abounded operator on a Hilbert space H with spectrum σ ( T ) contained in Γ,and there is a neighborhood U of Γ such that k ( T − λ ) − k ≤ dist( λ, Γ) − forall λ ∈ U \ Γ, then T is normal. Theorems of this type were first proved byNieminen [37] for the case Γ = R and by Donoghue [11] for the case when Γis a circle.If Γ is not smooth, such a result need no longer be true. A counterexamplecan be found in [46]. Even if Γ is a circle, the condition k ( T − λ ) − k ≤ C dist( λ, Γ) − , λ ∈ C \ Γ, where C is a constant greater than 1, is notsufficient for T to be similar to a normal operator; that is, for some invertible S and normal operator N , to have T = SN S − . See the paper Markus [30].Benamara and Nikolski [3, Section 3.2] have a general result in this direction,and in a related article [39], Nikolski and Treil give a counterexample where T is a rank one perturbation of a unitary operator with σ ( T ) ⊂ T .Nevertheless, the hypothesis in Stampfli’s theorem can be successfullyweakened. Denote by B ( H ) the set of bounded linear operators on a Hilbertspace H . We will prove the following: Theorem 1.
Let Γ ⊂ C be a C α Jordan curve, and Ω the domain itbounds. Let T ∈ B ( H ) be an operator with σ ( T ) ⊂ Γ . Assume that k ( T − λ ) − k ≤ λ, Γ) , λ ∈ U \ Ω , Date : March 12, 2018.2010
Mathematics Subject Classification.
Primary 47A10; Secondary 47B15, 47A60.The second author was supported by a grant from the Mathematics Department ofthe Universidad Aut´onoma de Madrid and the Project MTM2015-66157-C2-1-P of theMinistry of Economy and Competitiveness of Spain. This work forms part of his thesis,defended in 2017. The third author was supported by the Project MTM2015-66157-C2-1-P and by the ICMAT Severo Ochoa project SEV-2015-0554 of the Ministry of Economyand Competitiveness of Spain and the European Regional Development Fund (FEDER). for some open set U containing ∂ Ω , and k ( T − λ ) − k ≤ C dist( λ, Γ) , λ ∈ Ω , for some constant C . Then T is similar to a normal operator. In other words, we assume that a resolvent estimate with constant 1 issatisfied outside Ω and an estimate with constant C is satisfied inside Ω (then C ≥ C outside Ω; see the Remark atthe end of Section 4. The result gives a positive answer to Question 2 posedby Stampfli in [48], which he observed as being the case when Γ is a circle.The proof of Theorem 1 will use a generalization of a theorem of Putinarand Sandberg on complete K -spectral sets that was proved in [12]. In fact,this theorem is an easy corollary of this generalization and Lemma 6, whichis stated below. The connection of spectral sets and similarity problems wasalready observed by Stampfli in [48]. In Theorem 8 of that paper he provedvia different techniques a version of our Lemma 6 under the assumption thatΩ set is a spectral set for T rather than a K -spectral set, along with strongersmoothness conditions for the boundary of Ω.Many different kinds of conditions implying normality of an operator havebeen studied. See, for instance, [4] and the previous articles in this series.In [5], Campbell and Gellar studied operators T for which T ∗ T and T + T ∗ commute, showing, for instance, that if σ ( T ) is a subset of a verticalline or R , then T is normal. In [10] Djordjevi´c gave several conditionsfor an operator to be normal using the Moore-Penrose inverse. Gheondeaconsidered operators which are the product of two normal operators in [22].See also [32] and references therein.Here we exhibit conditions for an operator to be similar to a normaloperator in terms of estimates of (or more properly, bounds on) its resolvent.Others have done likewise. In [6], Van Casteren proved the following. Theorem VC1.
Let T ∈ B ( H ) be an operator with σ ( T ) ⊂ T . Assume that T satisfies the resolvent estimate k ( T − λ ) − k ≤ C (1 − | λ | ) − , | λ | < and k T n k ≤ C, n ≥ . Then T is similar to a unitary operator. An operator satisfying the last condition in this theorem is said to bepower bounded. Van Casteren improved this result in [7], as follows.
Theorem VC2.
Let T ∈ B ( H ) be an operator with σ ( T ) ⊂ T . Assume that T satisfies the resolvent estimate k ( T − λ ) − k ≤ C (1 − | λ | ) − , | λ | < , and for < r < and x ∈ H , Z | λ | = r k ( T − λ ) − x k | dλ | ≤ C k x k r − and Z | λ | = r k ( T ∗ − λ ) − x k | dλ | ≤ C k x k r − . Then T is similar to a unitary operator. ESOLVENT CRITERIA FOR SIMILARITY TO A NORMAL OPERATOR 3
By writing the power series for the resolvent, one can check that everypower bounded operator satisfies the last two conditions in this theorem.A related result was proved independently by Naboko in [33].
Theorem N.
Let T ∈ B ( H ) be an operator with σ ( T ) ⊂ T . Assume that T satisfies the resolvent conditions Z | λ | = r k ( T − λ ) − x k | dλ | ≤ C k x k r − , < r < , x ∈ H, and Z | λ | = r k ( T ∗ − λ ) − x k | dλ | ≤ C k x k − r , r < , x ∈ H. Then T is similar to a unitary operator. Each of Theorems VC1, VC2 and N in fact gives necessary and sufficientconditions for similarity to a unitary operator.In these theorems, it is possible to replace T by T ∗ , T − or T ∗− , obtainingyet other criteria. For related results and conditions, we refer the readerto [29], where some results close to Naboko’s were independently obtained,and to [38, Section 1.5.6]. The conditions in Theorems VC2 and N are notcomparable, in that there is no easy way of deducing one from the other.Additionally, we present analogues of criteria of Van Casteren and Naboko,generalized from the circle to a smooth curve Γ. The corresponding integralconditions use the existence of a family of curves, tending “nicely” to Γ (fromboth sides) in place of circles | λ | = r ; details are given at the beginning ofSection 5.The paper is organized as follows. Sections 2 and 3 are preparatory. Thefirst of these contains the basic facts about the pseudoanalytic extensionof functions and Dynkin’s functional calculus for an operator T with firstorder resolvent growth near the spectrum (that is, growth which is linear inthe resolvent). In Section 3, we use this calculus to show that the resolventestimates for an operator T with σ ( T ) ⊂ Γ are equivalent to correspondingresolvent estimates for η ( T ), where η is a smooth diffeomorphism from Γ to T , so that σ ( η ( T )) ⊂ T . Section 4 deals with the proof of Theorem 1, whilein Section 5, we formulate and prove analogues of mean-square criteria byVan Casteren and Naboko. Finally, Section 6 contains a brief discussion ofrelated results in the literature and a few examples.The authors are grateful to Maria Gamal’ for several insightful remarksand pointers to the literature.2. Dynkin’s functional calculus
Our key technical tool will be a generalization of the Riesz-Dunford func-tional calculus as defined by Dynkin in [13] using the Cauchy-Green formula.Before going into details, we need to set down some definitions and notation.Let Γ ⊂ C be a Jordan curve of class C α , 0 < α <
1. This means thatΓ is the image of T under a bijective map ψ : T → Γ such that ψ ∈ C ( T ), ψ ′ does not vanish and ψ ′ is H¨older α ; that is, | ψ ′ ( z ) − ψ ′ ( w ) | ≤ C | z − w | α , z, w ∈ T . ESOLVENT CRITERIA FOR SIMILARITY TO A NORMAL OPERATOR 4
A function f : Γ → C is said to belong to C α (Γ) if f ◦ ψ ∈ C ( T )and ( f ◦ ψ ) ′ is H¨older α . As an important example of such a function, take f = ψ − . This function has the additional properties that f (Γ) = T , f − exists as a map from T to Γ and is differentiable.The norm for f ∈ C α (Γ) is defined as k f k C α (Γ) = k f ◦ ψ k C ( T ) + k ( f ◦ ψ ) ′ k C ( T ) + k ( f ◦ ψ ) ′ k α , where k g k α = sup z,w ∈ T ,z = w | g ( z ) − g ( w ) || z − w | α . The definition of this norm depends on the choice of the parametrization ψ ,but different choices yield equivalent norms.Let T ∈ B ( H ) be an operator with σ ( T ) ⊂ Γ, where Γ is a Jordan curve ofclass C α . Assume that T satisfies the following resolvent growth condition:(1) k ( T − λ ) − k ≤ C dist( λ, Γ) , λ ∈ C \ Γ . Following Dynkin [13], a C α (Γ) functional calculus for T can be defined.Dynkin defines his calculus for a scale of function algebras including C α and operators satisfying other resolvent estimates (1). Only the case relevantto this paper is discussed here.To begin, recall the notion of pseudoanalytic extension. If f ∈ C α (Γ),then by [14, Theorem 2] there is a function F ∈ C ( C ) such that F | Γ = f and(2) (cid:12)(cid:12)(cid:12)(cid:12) ∂F∂z ( z ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C k f k C α (Γ) dist( z, Γ) α . Here, ∂∂z = (cid:16) ∂∂x + i ∂∂y (cid:17) and C is a constant depending only on Γ. Everysuch function F which extends f and satisfies (2) is called a pseudoanalyticextension of f .Dynkin uses the pseudoanalytic extension F to define the operator f ( T )by means of the Cauchy-Green integral formula. Let D be a domain withsmooth boundary such that Γ ⊂ D , and define f ( T ) = 12 πi Z ∂D F ( λ )( λ − T ) − dλ − π Z Z D ∂F∂z ( λ )( λ − T ) − dA ( λ ) . The inequality (2) for F and the resolvent estimate (1) for T imply thatthe second integral is well defined. It is possible to prove that the definitiondoes not depend on the particular choice of D or pseudoanalytic extension F .This calculus has the usual properties of a functional calculus: it is contin-uous from C α (Γ) to B ( H ), is linear and multiplicative, and coincides withthe natural definition of f ( T ) if f is rational. It also satisfies the spectralmapping property: σ ( f ( T )) = f ( σ ( T )).3. Passing from Γ to T We now explain how to use Dynkin’s functional calculus to pass from anoperator T with σ ( T ) ⊂ Γ to an operator A with σ ( A ) ⊂ T . The main resultof this section is Theorem 3, which relates the estimates for the resolvents ESOLVENT CRITERIA FOR SIMILARITY TO A NORMAL OPERATOR 5 of T and A . In this way, resolvent growth conditions for T imply equivalentconditions for A , and conversely. This equivalence plays a key role in whatfollows.The next lemma gives regularity conditions for a certain function η :Γ → T , which enables the construction of the operator A = η ( T ) using theDynkin functional calculus. Lemma 2.
Let Γ be a Jordan curve of class C α and η ∈ C α (Γ) afunction such that η (Γ) = T and η − : T → Γ exists and is differentiable.Fix any pseudoanalytic extension of η to C , also denoted by η . Then thereis a neighborhood U of Γ such that η : U → η ( U ) is a C diffeomorphism, η ( U ) is a neighborhood of T , and η is bi-Lipschitz in U ; that is, there arepositive constants c and C such that c | z − w | ≤ | η ( z ) − η ( w ) | ≤ C | z − w | , z, w ∈ U. A consequence of this lemma used frequently below is that for λ ∈ U ,dist( λ, Γ) and dist( η ( λ ) , T ) are comparable. Proof of Lemma 2.
Since ∂η/∂z ≡ η − : T → Γ is differentiable implies that the differential of η is non-singular on Γ.Therefore, for each point x ∈ Γ, there is an open ball B ( x, r ( x )) of center x and radius r ( x ) such that η is bi-Lipschitz on B ( x, r ( x )). By a compactnessargument, η is Lipschitz on some neighborhood of Γ.Pass to a finite collection { x j } on Γ such that the balls B ( x j , r ( x j ) / ε = min r ( x j ) /
2. Since η | Γ is injective, δ := min | x − y |≥ ε x,y ∈ Γ | η ( x ) − η ( y ) | > . It follows that there is some ρ > e δ := min | x − y |≥ ε dist( x, Γ) ≤ ρ, dist( y, Γ) ≤ ρ | η ( x ) − η ( y ) | > . Now check that η is bi-Lipschitz on the open set W = (cid:18) [ j B (cid:16) x j , r ( x j )2 (cid:17)(cid:19) ∩ { x ∈ C : dist( x, Γ) < ρ } . Given points x, y ∈ W , then either | x − y | < ε , so that x, y both belong tothe same ball B ( x k , r ( x k )), where η is bi-Lipschitz, or | x − y | ≥ ε . In thelatter case, | η ( x ) − η ( y ) | ≥ e δ ≥ e δ (cid:0) diam W (cid:1) − | x − y | . The injectivity of η follows from the bi-Lipschitz property. The fact thatit is possible to choose U ⊂ W so that η is a C diffeomorphism of U istrue because the differential of η is non-singular in some neighborhood ofΓ. Finally, since η (Γ) = T and η is an open mapping by being bi-Lipschitz, η ( U ) is an open neighborhood of T . (cid:3) The next theorem relates the resolvents estimates for T and η ( T ). ESOLVENT CRITERIA FOR SIMILARITY TO A NORMAL OPERATOR 6
Theorem 3.
Let Γ , η and U be as in Lemma 2 and T ∈ B ( H ) be an operatorsatisfying the resolvent estimate (1) . Let the operator η ( T ) be defined by the C α -functional calculus for T . Then σ ( η ( T )) ⊂ T , and for some C ≥ and depending on Γ , η , T , but not on λ or x , C − k ( T − λ ) − x k ≤ k ( η ( T ) − η ( λ )) − x k ≤ C k ( T − λ ) − x k , λ ∈ U \ Γ , x ∈ H. Proof.
The fact that σ ( η ( T )) ⊂ T follows from the spectral mapping theoremfor the Dynkin functional calculus.For λ ∈ U \ Γ, define functions ϕ λ , ψ λ ∈ C α (Γ) by(3) ϕ λ ( z ) = η ( z ) − η ( λ ) z − λ , ψ λ ( z ) = z − λη ( z ) − η ( λ ) . The operators ϕ λ ( T ) and ψ λ ( T ) are thus defined. In fact, ϕ λ ( T ) = ( η ( T ) − η ( λ ))( T − λ ) − , ψ λ ( T ) = ( T − λ )( η ( T ) − η ( λ )) − . Hence it suffices to show that k ϕ λ ( T ) k ≤ C , k ψ λ ( T ) k ≤ C , for C independent of λ .In fact, the functions ϕ λ and ψ λ are in U \ { λ } and since η is bi-Lipschitz, | ϕ λ ( z ) | ≤ C , | ψ λ ( z ) | ≤ C , z ∈ U \ { λ } . Let D be a domain with smooth boundary such that Γ ⊂ D ⊂ D ⊂ U and ε >
0, to be chosen later.By the Dynkin functional calculus, for λ ∈ D and for ε chosen smallenough so that B ( λ, ε ) ⊂ D , ϕ λ ( T ) = 12 πi Z ∂D ϕ λ ( z )( z − T ) − dz − πi Z ∂B ( λ,ε ) ϕ λ ( z )( z − T ) − dz − π Z Z D \ B ( λ,ε ) ∂ϕ λ ∂z ( z )( z − T ) − dA ( z ) . The case λ / ∈ D is similar.The norm k ϕ λ ( T ) k is bounded by estimating the three terms separately.For the second term, if ε < dist( λ, Γ), then Z ∂B ( λ,ε ) | ϕ λ ( z ) |k ( z − T ) − k | dz | ≤ C ε (dist( λ, Γ) − ε ) − . Letting ε →
0, it is seen that this term is negligible.The norm of the first term is bounded by12 π Z ∂D | ϕ λ ( z ) |k ( z − T ) − k | dz | ≤ C dist( ∂D, Γ) − length( ∂D ) . ESOLVENT CRITERIA FOR SIMILARITY TO A NORMAL OPERATOR 7
Finally, by using Lemma 2, the norm of the third term is bounded by
Z Z D \ B ( λ,ε ) (cid:12)(cid:12)(cid:12)(cid:12) ∂ϕ λ ∂z ( z ) (cid:12)(cid:12)(cid:12)(cid:12) k ( z − T ) − k dA ( z ) ≤ Z Z D | z − λ | (cid:12)(cid:12)(cid:12)(cid:12) ∂η∂z ( z ) (cid:12)(cid:12)(cid:12)(cid:12) k ( T − λ ) − k dA ( z ) ≤ C Z Z D | z − λ | − dist( z, Γ) α − dA ( z ) ≤ C Z Z D | η ( z ) − η ( λ ) | − dist( η ( z ) , T ) α − dA ( z ) ≤ C Z Z η ( D ) | ζ − η ( λ ) | − dist( ζ, T ) α − dA ( ζ ) ≤ C Z a ≤| ζ |≤ b | ζ − η ( λ ) | − | − | ζ || α − dA ( ζ ) . The change of variables ζ = η ( z ) has been performed and choice a < b is made so that the set η ( D ) is contained in the annulus a ≤ | ζ | ≤ b .By Lemma 4 below, the last term in this chain of inequalities is smallerthan a constant which is independent of λ . Thus k ϕ λ ( T ) k ≤ C , with C independent of λ .The proof that k ψ λ ( T ) k ≤ C is very similar, in this case using that (cid:12)(cid:12)(cid:12)(cid:12) ∂ψ λ ∂z ( z ) (cid:12)(cid:12)(cid:12)(cid:12) = | z − λ || η ( z ) − η ( λ ) | (cid:12)(cid:12)(cid:12)(cid:12) ∂η∂z ( z ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ C | η ( z ) − η ( λ ) | − (cid:12)(cid:12)(cid:12)(cid:12) ∂η∂z ( z ) (cid:12)(cid:12)(cid:12)(cid:12) . The remaining bounds are obtained in the same way. The proof is finished(modulo the next lemma). (cid:3)
Lemma 4.
Let < a < < b and a ≤ | w | ≤ b and − < β < . Then forsome C independent of w , Z Z a ≤| z |≤ b | z − w | − | − | z || β dA ( z ) ≤ C. Proof.
Performing a rotation if necessary, take w to be real and positive, sothat a ≤ w ≤ b . By passing to polar coordinates and using the inequality | re iθ − w | − ≤ C | r + iθ − w | − , which is valid for a ≤ r ≤ b , the integral in the statement of the lemma isless than a constant times Z Z [ a,b ] × [ − π,π ] | ζ − w | − | − Re ζ | β dA ( ζ ) . Now assume that a ≤ w ≤ ≤ w ≤ b will be similar).Estimate the integral by dividing the region of integration into two pieces. ESOLVENT CRITERIA FOR SIMILARITY TO A NORMAL OPERATOR 8
Put t = ( w + 1) /
2. Then
Z Z [ a,t ] × [ − π,π ] | − Re ζ | β | ζ − w | dA ( ζ ) + Z Z [ t,b ] × [ − π,π ] | − Re ζ | β | ζ − w | dA ( ζ ) ≤ Z Z [ a,t ] × [ − π,π ] | w − Re ζ | β | ζ − w | dA ( ζ ) + Z Z [ t,b ] × [ − π,π ] | − Re ζ | β | ζ − w | dA ( ζ ) ≤ Z Z [ a − w,t − w ] × [ − π,π ] | − Re ζ ′ | β | ζ ′ − w | dA ( ζ ′ ) + Z Z [ t,b ] × [ − π,π ] | − Re ζ | β | ζ − w | dA ( ζ ) ≤ Z Z [2 a − ,b ] × [ − π,π ] | − Re ζ | β | ζ − w | dA ( ζ ) , where we have performed the change of variables ζ ′ = ζ − w and usedthat 2 a − ≤ a − w and t − w ≤ b . By a change to polar coordinates ζ = 1 + re iθ , the single pole at 1 is seen to be of order strictly between − (cid:3) The proof of Theorem 1
We first recall the result from [12] to be used in the proof of the theorem.The statement given here is for a C α domain, although the original wasproved under weaker regularity conditions (see [12, Theorem 2]). Theorem A.
Let T ∈ B ( H ) and Ω a Jordan domain of class C α . Assumethere is some R > such that for every λ ∈ ∂ Ω there is some point µ k ( λ ) ∈ C \ Ω such that dist( µ k ( λ ) , ∂ Ω) = | µ k ( λ ) − λ | = R and k ( T − µ k ( λ )) − k ≤ R − . Then Ω is a complete K -spectral set for some K > . In other words, the conclusion is that there exists a constant K ≥ k f ( T ) k ≤ K k f k H ∞ (Ω) , for every (matrix-valued) rational function f with poles off of Ω (and hencefor every f which is continuous in Ω and analytic in Ω). This result affir-matively answers Question 3, posed by Stampfli in [48].A nice overview of complete K -spectral sets can be found in [40, Chap-ter 9]. A result of this property for an operator T is that T dilates to anoperator similar to a normal operator with spectrum in the boundary of thedomain. The additional assumptions in Theorem 1 will allow us to concludethat the operator T itself is similar to a normal operator. Curiously, this willrequire only knowing the weaker property that T has Ω as a K -spectral set;in other words, that k f ( T ) k ≤ K k f k H ∞ (Ω) only for scalar valued rationalfunctions with poles off of Ω. Lemma 5.
Under the hypotheses of Theorem 3, if η ( T ) is similar to aunitary operator, then T is is similar to a normal operator.Proof. Replacing T by ST S − , where S is such that Sη ( T ) S − = η ( ST S − )is unitary, it can be assumed that η ( T ) is unitary. Then ( η | Γ) − ∈ C α ( T ).Choose some β ∈ (0 , α ). Then ( η | Γ) − is in the class C β ( T ), which consistsof functions g ∈ C β ( T ) such that ( g ′ ( z ) − g ′ ( w )) / | z − w | β → z, w ∈ T , | z − w | →
0. Hence one can choose a sequence of rational functions { r n } ∞ n =1ESOLVENT CRITERIA FOR SIMILARITY TO A NORMAL OPERATOR 9 with poles off T such that r n tend to ( η | Γ) − in C β ( T ) (this follows,for instance, from [24, Theorem 2.12]). Thus r n ◦ η tend to the identityfunction in C β (Γ). By continuity of the C β (Γ)-functional calculus for T , ( r n ◦ η )( T ) tends to T in operator norm. The Dynkin functional calculusfor T is a homomorphism, and so ( r n ◦ η )( T ) = r n ( η ( T )). Since each r n ( η ( T ))is normal, it follows that T is also normal. (cid:3) As an alternative and a more direct proof of the last lemma, it seemstempting to argue that if A = η ( T ) is similar to a unitary operator, then η − ( A ) is defined, for instance, by the usual L ∞ -functional calculus for nor-mal operators, and η − ( A ) is similar to a normal operator. However, it isnot clear a priori why η − ( A ) = T .Theorem 1 is a straightforward consequence of Theorem A and the fol-lowing lemma. Lemma 6.
Let Γ ⊂ C be a C α Jordan curve, and Ω the domain it bounds.Let T ∈ B ( H ) be an operator with σ ( T ) ⊂ Γ . Assume that Ω is a K -spectralset for T and k ( T − λ ) − k ≤ C dist( λ, Γ) , λ ∈ Ω , for some constant C > . Then T is similar to a normal operator.Proof. Let η : Ω → D be the Riemann map. Since ∂ Ω is of class C α ,then η ∈ C α ( ∂ Ω) (see, for instance, [43, Theorem 3.6]). Extend η pseu-doanalytically to C \ Ω. Now as η satisfies its assumptions, we can applyLemma 2.Because | η n | ≤ n ≥
0, and Ω is K -spectral for T , the operator η ( T ) is power bounded. By Theorem 3, and the fact that dist( λ, ∂ Ω) anddist( η ( λ ) , T ) are comparable, k ( η ( T ) − λ ) − k ≤ C − | λ | , | λ | < . Applying Theorem VC1 it follows that η ( T ) is similar to a unitary operator,and so by Lemma 5, T is similar to a normal operator. (cid:3) Proof of Theorem 1.
Theorem A implies that Ω is a complete K -spectral setfor T . It suffices to apply Lemma 6. (cid:3) Remarks.
It is straightforward to deduce an analogous result assuming anestimate with constant 1 inside the domain Ω and an estimate with a con-stant C outside the domain. Indeed, put R = ( T − z ) − , for some fixed z ∈ Ω. It follows from [12, Lemma 7] that if k ( T − λ ) − k ≤ dist( λ, Γ) − ,then k ( R − µ ) − k ≤ dist( µ, e Γ) − , where µ = ( λ − z ) − and e Γ is the image ofΓ under the map z ( z − z ) − . Writing the resolvent of R in terms of theresolvent of T , it is also easy to obtain an estimate for R with a constant C ′ > e Γ. Since the map z ( z − z ) − sends the inside of Γ onto the outside of e Γ and vice versa, it suffices to applyTheorem 1 to R .The conclusion of Lemma 6 is that T is similar to a normal operator, andso the set Ω (and even the boundary of Ω) must in fact be a complete K ′ -spectral set for T for some K ′ >
1. As it follows from the celebrated example
ESOLVENT CRITERIA FOR SIMILARITY TO A NORMAL OPERATOR 10 of Pisier, combined with the main result of [42], there exists an operator T on H with σ ( T ) ⊂ T , which is polynomially bounded, but not completelybounded. So, in Lemma 6 one cannot replace the resolvent estimate inside Ωjust by the condition that σ ( T ) ⊂ ∂ Ω. Recently, Gamal’ [21] has constructedseveral new examples of operators that are polynomially bounded but notcompletely polynomially bounded. In particular, any operator given in [21,Corollary 2.8] is quasisimilar to an absolute continuous unitary operator U and also satisfies σ ( T ) ⊂ T (the latter follows from [20, Theorem 2.4]).Using the techniques outlined here, this counterexample can be transferredfrom D to other sets Ω.5. Mean-square type resolvent estimates
In this section we give criteria for similarity to a normal operator analo-gous to the results by Van Casteren [7] and Naboko [33] in the context of C α Jordan curves. First of all, a substitute for the curves r T is needed.To this end, we give the following definition. Definition.
Let Γ ⊂ C be a Jordan curve and Ω the region it bounds. Afamily of Jordan curves { γ s }
0, length( γ s ∩ B ( x, r )) ≤ Cr .The family { γ s } γ ± s ⊂ U for every 0 < s ≤
1. The curves { γ + s } tend nicely to Γ fromoutside and the curves { γ − s } tend nicely to Γ from inside.It will be proved that the mean-square type resolvent estimates consideredhere do not depend on the concrete choice of the family of curves { γ s } tending nicely to Γ. This will follow from a lemma concerning Smirnovspaces.Recall that the Smirnov space E (Ω , H ) of H -valued function on a (nice)domain Ω is defined as the L ( ∂ Ω)-closure of the H -valued rational functionswith poles off Ω. The following lemma dates back to David and his theoremon the boundedness of certain singular integral operators on Ahlfors regularcurves. In particular, it follows from the results in [9, Proposition 6]. Lemma 7.
Let Ω , Ω be Jordan domains with Ahlfors regular boundariessuch that Ω ⊂ Ω . If H is a Hilbert space and f ∈ E (Ω , H ) , then f | Ω ∈ E (Ω , H ) and k f | Ω k E (Ω ,H ) ≤ C k f k E (Ω ,H ) , ESOLVENT CRITERIA FOR SIMILARITY TO A NORMAL OPERATOR 11 for some constant C depending only on the Ahlfors constants for ∂ Ω and ∂ Ω . Lemma 8.
Let Γ be a Jordan curve, T ∈ B ( H ) with σ ( T ) ⊂ Γ , and { γ s }
Let Γ ⊂ C be a Jordan curve of class C α , Ω the region itbounds and T ∈ B ( H ) with σ ( T ) ⊂ Γ . If (4) Z γ s k ( T − λ ) − x k | dλ | ≤ C k x k s , x ∈ H, < s ≤ , ESOLVENT CRITERIA FOR SIMILARITY TO A NORMAL OPERATOR 12 for some constant C independent of x and s and some family of curves { γ s } which tends nicely to Γ from the inside ( respectively, outside ) , then k ( T − λ ) − k ≤ C ′ dist( λ, Γ) , λ ∈ Ω ( respectively, λ ∈ C \ Ω) , for some constant C ′ independent of λ .Proof. Assume that { γ s } tends nicely to Γ from the inside. Let η be afunction as in the statement of Lemma 2, and U the neighborhood of Γ thatappears in that lemma. Fix λ ∈ Ω ∩ U . Then because η is bi-Lipschitz, t = dist( λ, Γ) is comparable to dist( η ( λ ) , T ). Put r = 1 − dist( η ( λ ) , T ) / η − ( r T ). This lies inside Ω anddist( z, Γ) is comparable to t for every z ∈ Λ. Therefore, it is possible tochoose 0 < s ≤ t ≤ C s , and Λ is inside the region bounded by γ s .Fix x ∈ H and put f ( z ) = ( η ( T ) − z ) − x , and g ( z ) = f ( z/r ). By theusual pointwise estimate for a function in H , the Hardy space of the disk, k g ( z ) k ≤ (1 − | z | ) − / k g k E ( D ,H ) = (1 − | z | ) − / k f k E ( r D ,H ) ≤ (1 − | z | ) − / k f k E ( W,H ) , where W is the domain bounded by η ( γ s ) and the last inequality comes fromLemma 7. Now by Theorem 3 and (4), k f k E ( W,H ) = Z η ( γ s ) k ( η ( T ) − z ) − x k | dz |≤ C Z γ s k ( η ( T ) − η ( w )) − x k | dw | ≤ C Z γ s k ( T − w ) − x k | dw | ≤ C k x k s . Hence, k ( η ( T ) − z/r ) − x k ≤ C (1 − | z | ) − s − k x k , and the inequality above is valid for all x ∈ H . Putting z = rη ( λ ) yields k ( η ( T ) − η ( λ )) − k ≤ C (1 − | rη ( λ ) | ) − s − ≤ C t − . By another application of Theorem 3, k ( T − λ ) − k ≤ C ′ t = C ′ dist( λ, Γ) . The case when { γ s } tends nicely to Γ from the outside is proved by ap-plying the inversion z ( z − z ) − , as in the proof of Lemma 8. (cid:3) We now state and prove generalizations of Theorems VC2 and N in The-orems 10 and Theorem 12. The proofs both follow the same line of rea-soning, using the tools so far developed to pass to T and then applyingVan Casteren’s or Naboko’s theorem. It is worth highlighting that, as withthe original theorems, there is no easy way to deduce either result from theother. Theorem 10 (Van Casteren-type theorem for curves) . Let Γ ⊂ C be aJordan curve of class C α , Ω the region it bounds and T ∈ B ( H ) with σ ( T ) ⊂ Γ . Let { γ s }
First, assume that T satisfies the three resolvent conditions. Let η : U → C be a function as in the statement of Lemma 2. Take γ s ⊂ U for every 0 < s ≤
1. By Lemma 9, the operator T satisfies the resolventestimate (1), so η ( T ) is defined by the C α (Γ)-functional calculus for T .By Theorem 3 and the fact that dist( λ, Γ) and dist( η ( λ ) , T ) are comparable, k ( η ( T ) − λ ) − k ≤ C − | λ | , | λ | < , as well as Z η ( γ s ) k ( η ( T ) − λ ) − x k | dλ | ≤ C k x k s , x ∈ H, < s ≤ , which follows by making a change of variables λ = η ( µ ) and applying The-orem 3.Since η : U → η ( U ) is a C diffeomorphism and bi-Lipschitz, the familyof curves { η ( γ s ) }
0, it follows that Z r T k ( η ( T ) − λ ) − x k | dλ | ≤ C k x k r − , < r < , x ∈ H. ESOLVENT CRITERIA FOR SIMILARITY TO A NORMAL OPERATOR 14
By Theorem 3 (although since T is normal, a simpler argument could bedevised), for some constant C > U of Γ, C − k ( T − λ ) − x k ≤ k ( η ( T ) − η ( λ )) − x k ≤ C k ( T − λ ) − x k , λ ∈ U \ Γ , x ∈ H, Therefore, Z η − ( r T ) k ( T − λ ) − x k | dλ | ≤ C k x k r − , < r < , x ∈ H. The family of curves e γ s = η − ((1 + s ) T ) tends nicely to Γ. Apply Lemma 8to get that T satisfies (5). Use of similar reasoning, but with T ∗ instead of T and e η instead of η yields the inequality (6). (cid:3) Sz.-Nagy proved in [49] that an operator T is similar to a unitary operatorif and only if k T n k ≤ C for all n ∈ Z . It is easy to use this result to showthat if σ ( T ) ⊂ Γ, the resolvent estimate (1) holds and k η n ( T ) k ≤ C for all n ∈ Z , then T is similar to a normal operator.The following corollary is a generalization of Theorem VC1. Note thathere it is only assumed that k η ( T ) n k ≤ C for all n ≥
0. The proof issimilar to the proof of Theorem 10, but Theorem VC1 is used instead ofTheorem VC2.
Corollary 11.
Let Γ ⊂ C be a Jordan curve of class C α , and T ∈ B ( H ) with σ ( T ) ⊂ Γ . Assume that k ( T − λ ) − k ≤ C dist( λ, Γ) , λ ∈ C \ Γ . Let η : Γ → T be a function as in the statement of Lemma 2. Define theoperator η ( T ) by the C α -functional calculus. If η ( T ) is power bounded,then T is similar to a normal operator. Theorem 12 (Naboko-type theorem for curves) . Let Γ ⊂ C be a Jordancurve of class C α , Ω the region it bounds and T ∈ B ( H ) with σ ( T ) ⊂ Γ .Let { γ s }
The proof of this theorem is like the proof of Theorem 10. If T satisfies the two conditions in the statement of this theorem, instead ofusing Van Casteren’s theorem, use Naboko’s Theorem N to show that η ( T )is similar to a unitary operator. We only sketch the proof.First, Lemma 9 implies that T satisfies the resolvent estimate (1). Choosea function η as in Lemma 2. The operator η ( T ) is well defined. By Theo-rem 3 and Lemma 8, Z | λ | = r k ( η ( T ) − λ ) − x k | dλ | ≤ C k x k r − , x ∈ H, < r < ESOLVENT CRITERIA FOR SIMILARITY TO A NORMAL OPERATOR 15 and Z | λ | = r k ( η ( T ) ∗ − λ ) − x k | dλ | ≤ C k x k − r , x ∈ H, < r < . Theorem N, then gives that η ( T ) is similar to a unitary operator. It followsthat T is similar to a normal operator by Lemma 5.The converse direction is proved as in Theorem 10. (cid:3) Some comments and examples
So far we have only discussed operators with spectrum on some smoothcurve in C . Similar results can be presented if the spectrum is allowedto be a union of a smooth curve and a sequence of points tending to thiscurve. In [3], Benamara and Nikolski show that a contraction T with fi-nite defects is similar to a normal operator if and only if σ ( T ) = D and k ( T − λ ) − k ≤ C dist( λ, σ ( T )) − for all λ ∈ C \ σ ( T ). For such a contrac-tion, σ ( T ) \ T is always a Blaschke sequence in D . Moreover, Benamaraand Nikolski prove that the resolvent estimate forces σ ( T ) ∩ D to be quitesparse (more precisely, it has to satisfy the ∆-Carleson condition). Laterin [25], Kupin studied contractions with infinite defects. He proved thatif the spectrum of a contraction T is not all D , and if it satisfies both k ( T − λ ) − k ≤ C dist( λ, σ ( T )) − for all λ ∈ σ ( T ) and the so-called UniformTrace Boundedness condition, then it is similar to a normal operator. Ananalogue of Uniform Trace Boundedness condition for dissipative operatorswas given by Vasyunin and Kupin in [54], and then applied to integral op-erators. In [26], Kupin also uses the Uniform Trace Boundedness conditionto give conditions for an operator similar to a contraction to be similar to anormal operator.On the other hand, Kupin and Treil showed in [27] that if T is a con-traction with σ ( T ) = D and k ( T − λ ) − k ≤ C dist( λ, σ ( T )) − but one onlyassumes that I − T ∗ T is trace class (instead of finite rank), then T need notbe similar to a normal operator, thus solving a conjecture in [3].All these results concern operators with thin spectrum (in other words,those for which the area of the spectrum is zero). For operators having thick spectrum (so non-zero area), in general there is no hope of obtaining criteriafor similarity to a normal operator solely in terms of resolvent operator es-timates. Indeed, for any hyponormal operator T , the best possible estimate k ( T − λ ) − k = dist( λ, σ ( T )) − holds, and for any compact set F of positivearea there exists a hyponormal operator T not similar to a normal one with σ ( T ) = F .Resolvent conditions for similarity to other classes of operators, such asselfadjoint operators or isometries, have also been considered in the litera-ture. Faddeev gives conditions in [16] for similarity to an isometry in thecase where dim ker( T ∗ − λI ) = 1 for all λ ∈ D . In [44], Popescu also statesseveral conditions for similarity to an isometry.In [28], Malamud gives a series of abstract conditions for an operator A to be similar to a selfadjoint one. His conditions involve, in particular,resolvent estimates of the form k V / ( A − λ ) − k ≤ C | Im λ | − / , where V = | Im A | . These estimates are related to Theorem N. He applies his results to ESOLVENT CRITERIA FOR SIMILARITY TO A NORMAL OPERATOR 16 a triangular operator on L ([0 , , dµ ) of the form(7) ( Af )( x ) = α ( x ) f ( x ) + i Z x K ( x, t ) f ( t ) dµ ( t ) , for an Hermitian kernel K ( x, t ).Naboko and Tretter used Theorem N in [36] to examine operators of theform (7), where K ( x, t ) = φ ( x ) ψ ( t ) (so that K ( x, t ) need not be Hermitian)and φψ ≡
0. By using our criteria for curves, most likely Naboko andTretter’s results can be extended to analogous operators on L of a curve,though it might be rather technical.Also concretely, such conditions have been used in the study of differentialoperators. For example, in [17], Faddeev and Shterenberg use a version ofTheorem N in examining similarity to a selfadjoint operator for operatorsof the form A = − sign x | x | α p ( x ) d dx , where α > − p is a positive functionwhich is bounded above and below. Their criteria were further generalizedby Karabash, Kostenko and Malamud in [23].Resolvent growth conditions for similarity to a unitary operator have alsobeen used in the study of Toeplitz operators with unimodular symbol; see [8,18, 41] and references therein.Article [15] contains a discussion of the relationship between the growthof powers of an operator T with σ ( T ) ⊂ D , first order growth of its resolventoutside D and the size of the set σ ( T ) ∩ T .The conditions for a contraction T to be similar to a unitary operator interms of the characteristic function of T are well known. Given a contraction T ∈ B ( H ), one defines defect operators D T = ( I − T ∗ T ) , D T ∗ = ( I − T T ∗ ) and defect spaces D T = D T H , D T ∗ = D ∗ T H . For λ ∈ D , the characteristicfunction Θ T ( λ ) : D T → D T ∗ is given byΘ T ( λ ) = [ − T + λD T ∗ ( I − λT ∗ ) − D T ] | D T . As Sz.-Nagy and Foias proved in [50] (see also Sz.-Nagy and Foias [51, Chap-ter 9]), T is similar to a unitary operator if and only if Θ T ( λ ) is invertiblefor all λ ∈ D and(8) sup λ ∈ D k Θ T ( λ ) − k < ∞ . L.A. Saknovich extended the results of Sz.-Nagy and Foias to operatorswhich are not necessarily contractions in [45]. Saknovich’s condition is onlysufficient for similarity to a unitary operator and not necessary in general.See also Naboko [35, Theorem 12].In a series of articles, Naboko constructed and studied a functional modelfor non-dissipative perturbations of self-adjoint operators. A detailed ex-position of this model can be found in [34]. In that paper, the problem ofexistence of wave operators in this context is discussed. A functional modelfor perturbations of normal operators with spectrum on a curve, extendingNaboko’s model, has been developed by Tikhonov in [52] and subsequentpapers.
Example . A purely contractive function Θ can be chosen satisfying thecondition (8) and the Sz.-Nagy Foias model used to construct a completely
ESOLVENT CRITERIA FOR SIMILARITY TO A NORMAL OPERATOR 17 non-unitary contraction T such that Θ T = Θ. Such a contraction is non-unitary and similar to a unitary operator, and so σ ( T ) ⊂ T . Hence k ( T − λ ) − k ≤ C − | λ | , | λ | < . Since T is also a contraction, by von Neumann’s inequality k ( T − λ ) − k ≤ | λ | − , | λ | > . Recall that, by Stampfli’s theorem stated in the introduction, if under theseconditions T satisfies k ( T − λ ) − k ≤ || λ | − | , | λ | 6 = 1 , then T must be normal. Thus this yields an example of an operator whichsatisfies the hypotheses of Theorem 1 and is non-normal.There are also examples of this type among the class of ρ -contractions.If ρ >
0, an operator T ∈ B ( H ) is called a ρ -contraction if there is a largerHilbert space K ⊃ H and a unitary U ∈ B ( K ) such that T n = ρP H U n | H, n = 1 , , . . . , where P H denotes the orthogonal projection onto H . The classes C ρ of ρ -contractions are nested and increasing with ρ , and the class C coincideswith the class of contractions.If T is a ρ -contraction with ρ ≥
2, then k ( T − λ ) − k ≤ | λ | − , < | λ | < ρ − ρ − . (Here ρ − ρ − = + ∞ if ρ = 2.) Therefore, any ρ -contraction which is similarto a unitary operator also satisfies the hypotheses of Theorem 1. If T isa 2-contraction, then one may take the set U = C in the hypotheses ofTheorem 1. However, for a ρ -contraction with ρ > U will in general be asmaller set.It is natural to ask if there is an example of a ρ -contraction which isnot a contraction and where the spectrum is contained in the unit circle.Stampfli shows that this can occur in [48, Example 2]. There, ρ = 2 andthe spectrum of the operator is a single point. He proves that since thespectrum is countable, this operator must be normal.Another example, this time with a bilateral weighed shift, is given below.Here the spectrum is the whole unit circle, and the operator is similar to aunitary, but is not normal.Any such example must have uncountable spectrum. Consequently, itwould be interesting to know for a non-normal operator T with spectrum σ ( T ) contained in a curve Γ and satisfying the hypotheses of Theorem 1 (sothat it is similar to a normal operator), just how small σ ( T ) can be. See[15] and references therein for a discussion of some similar questions. Example . Assume that α, β >
0, max( α, β ) >
1, and α + β ≤
4. Let T bethe bilateral weighted shift T on ℓ ( Z ) with weights { . . . , , , α , β, , , . . . } , ESOLVENT CRITERIA FOR SIMILARITY TO A NORMAL OPERATOR 18 defined by T ( . . . , x − , x , x , x , . . . ) = ( . . . , x − , x − , αx , βx , x , . . . ) . (Here indicates the 0-th component). Then T is a 2-contraction which isnot a contraction, yet is similar to a unitary operator.Obviously, k T k = max( α, β, >
1. Since α, β >
0, the operator T issimilar to the unitary bilateral shift U on ℓ ( Z ) with all weights equal to 1.It remains to show that T is a 2-contraction.Recall that T is a 2-contraction if and only ifRe( θT ) ≤ I, | θ | = 1 . Since θT is unitarily equivalent to T when | θ | = 1, it is enough to check thisinequality for θ = 1.Put A = 2 Re T , and check that σ ( A ) ∩ (2 , + ∞ ) = ∅ . Since A is a finiterank perturbation of U + U ∗ and σ ( U + U ∗ ) = [ − , A has no eigenvalues in (2 , + ∞ ). Assume that x = ( x n ) n ∈ Z is a non-zerovector in ℓ ( Z ) that satisfies ( A − λ ) x = 0 for some λ >
2. This means that(9) x n − λx n +1 + x n +2 = 0 , | n | ≥ , (10) x − − λx + αx = 0 , (11) αx − λx + βx = 0 , (12) βx − λx + x = 0 . Put u ± = u ± ( λ ) = λ ± √ λ − . Then (9) and x ∈ ℓ ( Z ) imply that for some non-zero a, b , x n = au n − , n ≥ ,x n = bu n + , n ≤ . The quotients y n = x n +1 x n satisfy y n = u − , n ≥ ,y n = u + , n ≤ − . If F x n − λx n +1 + Gx n +2 = 0 , then y n is obtained from y n +1 by applying the M¨obius transformation z F − Gz + λ , which can be encoded by the 2 × (cid:18) F − G λ (cid:19) . The compositionof M¨obius transformations reduces to multiplying the corresponding 2 × u + ( λ ) = y − = M ( λ ) y + M ( λ ) M ( λ ) y + M ( λ ) = M ( λ ) u − ( λ ) + M ( λ ) M ( λ ) u − ( λ ) + M ( λ ) , where (cid:18) M ( λ ) M ( λ ) M ( λ ) M ( λ ) (cid:19) = (cid:18) − α λ (cid:19) (cid:18) α − β λ (cid:19) (cid:18) β − λ (cid:19) . ESOLVENT CRITERIA FOR SIMILARITY TO A NORMAL OPERATOR 19
Putting f ( λ ) = u + ( λ ) (cid:0) M ( λ ) u − ( λ ) + M ( λ ) (cid:1) − (cid:0) M ( λ ) u − ( λ ) + M ( λ ) (cid:1) == λ [( λ − ( α + β )) u + ( λ ) + u − ( λ )] , it follows that f ( λ ) = 0. However, since λ > α + β ≤
4, and u + ( λ ) and u − ( λ ) are positive, f ( λ ) >
0. This is a contradiction.In the above argument, the numerical radius of the weighted shift T wascomputed by examining the spectrum of its real part. There are severalworks in the literature devoted to the study of the numerical radius ofweighted shifts, using similar techniques. See [53] and references therein.In [2], Andˆo and Takahashi proved that if an operator T is polynomiallybounded and there exist an injective operator X and a unitary operator W with non-singular spectral measure with respect to the Lebesgue measureon T , and such that XT = W X , then T is similar to a unitary operator.Moreover, if such T is also a ρ -contraction for some ρ >
0, then T is itselfunitary. This does not apply in Example 2, since the operator T is similar tothe bilateral shift in L ( T ), the spectral measure of which is not singular. Asimilar result is contained in Mlak [31]. See Gamal’ [19] and the referencestherein for extensions of these results. Example . One can easily construct non-normal operators which satisfythe hypotheses of Theorem 1 for a Jordan domain Ω = D . Let A be a non-unitary contraction which is similar to a unitary operator. Take a Riemannmapping ϕ : Ω → D and put ψ = ϕ − . The operator T = ϕ ( A ) is welldefined and non-normal. If λ ∈ C \ Ω, then by von Neumann’s inequality k ( T − λ ) − k ≤ k ( ϕ − λ ) − k H ∞ ( D ) = dist( λ, Ω) − . If λ ∈ Ω, the inequality k ( T − λ ) − k ≤ C dist( λ, Ω) − follows from the fact that T is similar to a normal operator. The operator T satisfies the hypotheses of Theorem 1.It is not obvious how to use a Riemann mapping in a similar manner toget a result analogous to Example 2 for a general Jordan domain Ω. References [1] L. Ahlfors,
Zur Theorie der ¨Uberlagerungsfl¨achen , Acta Math. (1935), no. 1, 157–194.[2] T. Ando and K. Takahashi, On operators with unitary ρ -dilations , Ann. Polon. Math. (1997), 11–14. Volume dedicated to the memory of W lodzimierz Mlak.[3] N.-E. Benamara and N. Nikolski, Resolvent tests for similarity to a normal operator ,Proc. London Math. Soc. (3) (1999), no. 3, 585–626.[4] S. K. Berberian, Some conditions on an operator implying normality. III , Proc. JapanAcad. (1970), 630–632.[5] S. L. Campbell and R. Gellar, Linear operators for which T ∗ T and T + T ∗ commute.II , Trans. Amer. Math. Soc. (1977), 305–319.[6] J. A. Van Casteren, A problem of Sz.-Nagy , Acta Sci. Math. (Szeged) (1980),no. 1-2, 189–194.[7] , Operators similar to unitary or selfadjoint ones , Pacific J. Math. (1983),no. 1, 241–255.
ESOLVENT CRITERIA FOR SIMILARITY TO A NORMAL OPERATOR 20 [8] D. N. Clark,
On Toeplitz operators with unimodular symbols , Operators in indefinitemetric spaces, scattering theory and other topics (Bucharest, 1985), Oper. TheoryAdv. Appl., vol. 24, Birkh¨auser, Basel, 1987, pp. 59–68.[9] G. David,
Op´erateurs int´egraux singuliers sur certaines courbes du plan complexe ,Ann. Sci. ´Ecole Norm. Sup. (4) (1984), no. 1, 157–189.[10] D. S. Djordjevi´c, Characterizations of normal, hyponormal and EP operators , J. Math.Anal. Appl. (2007), no. 2, 1181–1190.[11] W. F. Donoghue Jr.,
On a problem of Nieminen , Inst. Hautes ´Etudes Sci. Publ. Math. (1963), 31–33.[12] M. A. Dritschel, D. Est´evez, and D. Yakubovich, Tests for complete K -spectral sets ,J. Funct. Anal. (2017), no. 3, 984–1019.[13] E. M. Dyn ′ kin, An operator calculus based on the Cauchy-Green formula , Zap. Nauˇcn.Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) (1972), 33–39 (Russian); Eng-lish transl., J. Soviet Math. (1975), no. 4, 329–334.[14] , Pseudoanalytic continuation of smooth functions. Uniform scale , Mathemat-ical programming and related questions (Proc. Seventh Winter School, Drogobych,1974), Central `Ekonom.-Mat. Inst. Akad. Nauk SSSR, Moscow, 1976, pp. 40–73.[15] O. El-Fallah and T. Ransford,
Peripheral point spectrum and growth of powers ofoperators , J. Operator Theory (2004), no. 1, 89–101.[16] M. M. Faddeev, Similarity of an operator to an isometric operator , Funktsional. Anal.i Prilozhen. (1989), no. 2, 77–78; English transl., Funct. Anal. Appl. (1989),no. 2, 149–151.[17] M. M. Faddeev and R. G. Shterenberg, On the similarity of some differential operatorsto selfadjoint operators , Mat. Zametki (2002), no. 2, 292–302; English transl.,Math. Notes (2002), no. 1-2, 261–270.[18] M. F. Gamal’, On Toeplitz operators similar to isometries , J. Operator Theory (2008), no. 1, 3–28.[19] , On power bounded operators that are quasiaffine transforms of singular uni-taries , Acta Sci. Math. (2011), no. 3-4, 589–606.[20] , On quasisimilarity of polynomially bounded operators , Acta Sci. Math.(Szeged) (2015), no. 1-2, 241–249.[21] , Examples of cyclic polynomially bounded operators that are not similar tocontractions , Acta Sci. Math. (Szeged) (2016), no. 3-4, 597–628.[22] A. Gheondea, When are the products of normal operators normal? , Bull. Math. Soc.Sci. Math. Roumanie (N.S.) (2009), no. 2, 129–150.[23] I. M. Karabash, A. S. Kostenko, and M. M. Malamud,
The similarity problem for J -nonnegative Sturm-Liouville operators , J. Differential Equations (2009), no. 3,964–997.[24] Y. Katznelson, An introduction to harmonic analysis , John Wiley & Sons, Inc., NewYork-London-Sydney, 1968.[25] S. Kupin,
Linear resolvent growth test for similarity of a weak contraction to a normaloperator , Ark. Mat. (2001), no. 1, 95–119.[26] S. Kupin, Operators similar to contractions and their similarity to a normal operator ,Indiana Univ. Math. J. (2003), no. 3, 753–768.[27] S. Kupin and S. Treil, Linear resolvent growth of a weak contraction does not implyits similarity to a normal operator , Illinois J. Math. (2001), no. 1, 229–242.[28] M. M. Malamud, On the similarity of a triangular operator to a diagonal operator ,Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) (2000),201–241, 367 (Russian); English transl., J. Math. Sci. (N. Y.) (2003), no. 2,2199–2222.[29] ,
A criterion for a closed operator to be similar to a selfadjoint operator ,Ukrain. Mat. Zh. (1985), no. 1, 49–56, 134 (Russian); English transl., Ukr. Math.J. (1985), no. 1, 41–48.[30] A. S. Markus, Some tests of completeness of the system of root vectors of a linearoperator and the summability of series in that system , Dokl. Akad. Nauk SSSR (1964), 753–756 (Russian).
ESOLVENT CRITERIA FOR SIMILARITY TO A NORMAL OPERATOR 21 [31] W. Mlak,
Algebraic polynomially bounded operators , Ann. Polon. Math. (1974),133–139. Collection of articles dedicated to the memory of Tadeusz Wa˙zewski, II.[32] M. S. Moslehian and S. M. S. Nabavi Sales, Some conditions implying normality ofoperators , C. R. Math. Acad. Sci. Paris (2011), no. 5-6, 251–254.[33] S. N. Naboko,
Conditions for similarity to unitary and selfadjoint operators , Funkt-sional. Anal. i Prilozhen. (1984), no. 1, 16–27.[34] , Functional model of perturbation theory and its applications to scattering the-ory , Proc. Steklov Inst. of Mathematics (1981), 85–116. Boundary value problemsof mathematical physics and related questions in the theory of functions, 10.[35] ,
On the singular spectrum of a nonselfadjoint operator , Zap. Nauchn. Sem.Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) (1981), 149–177, 266 (Russian,with English summary); English transl., J. Math. Sci. (N. Y.) (1983), no. 6, 1793–1813.[36] S. N. Naboko and C. Tretter, Lyapunov stability of a perturbed multiplication operator ,Contributions to operator theory in spaces with an indefinite metric (Vienna, 1995),Oper. Theory Adv. Appl., vol. 106, Birkh¨auser, Basel, 1998, pp. 309–326.[37] T. Nieminen,
A condition for the self-adjointness of a linear operator , Ann. Acad.Sci. Fenn. Ser. A I No (1962), 5.[38] N. K. Nikolski,
Operators, functions, and systems: an easy reading. Vol. 2 , Mathemat-ical Surveys and Monographs, vol. 93, American Mathematical Society, Providence,RI, 2002.[39] N. Nikolski and S. Treil,
Linear resolvent growth of rank one perturbation of a unitaryoperator does not imply its similarity to a normal operator , J. Anal. Math. (2002),415–431.[40] V. Paulsen, Completely bounded maps and operator algebras , Cambridge Studies inAdvanced Mathematics, vol. 78, Cambridge University Press, Cambridge, 2002.[41] V. V. Peller,
Spectrum, similarity, and invariant subspaces of Toeplitz operators , Izv.Akad. Nauk SSSR Ser. Mat. (1986), no. 4, 776–787, 878 (Russian).[42] S. Petrovi´c, A dilation theory for polynomially bounded operators , J. Funct. Anal. (1992), no. 2, 458–469.[43] Ch. Pommerenke,
Boundary behaviour of conformal maps , Grundlehren der Math-ematischen Wissenschaften [Fundamental Principles of Mathematical Sciences],vol. 299, Springer-Verlag, Berlin, 1992.[44] G. Popescu,
On similarity of operators to isometries , Michigan Math. J. (1992),no. 3, 385–393.[45] L. A. Sakhnovich, Nonunitary operators with absolutely continuous spectrum , IzvestiaAkad. Nauk SSSR, Ser. Mat. (1969), no. 1, 52–64.[46] J. G. Stampfli, Hyponormal operators and spectral density , Trans. Amer. Math. Soc. (1965), 469–476.[47] ,
A local spectral theory for operators , J. Functional Analysis (1969), 1–10.[48] , A local spectral theory for operators. III. Resolvents, spectral sets and simi-larity , Trans. Amer. Math. Soc. (1972), 133–151.[49] B. de Sz. Nagy,
On uniformly bounded linear transformations in Hilbert space , ActaUniv. Szeged. Sect. Sci. Math. (1947), 152–157.[50] B. Sz.-Nagy and C. Foia¸s, Sur les contractions de l’espace de Hilbert. X. Contractionssimilaires `a des transformations unitaires , Acta Sci. Math. (Szeged) (1965), 79–91(French).[51] B. Sz.-Nagy and C. Foia , s, Harmonic analysis of operators on Hilbert space , Translatedfrom the French and revised, North-Holland Publishing Co., Amsterdam-London;American Elsevier Publishing Co., Inc., New York; Akad´emiai Kiad´o, Budapest, 1970.[52] A. S. Tikhonov,
A functional model and duality of spectral components for operatorswith a continuous spectrum on a curve , Algebra i Analiz (2002), no. 4, 158–195(Russian); English transl., St. Petersburg Math. J. (2003), no. 4, 655–682.[53] B. Undrakh, H. Nakazato, A. Vandanjav, and M.-T. Chien, The numerical radius ofa weighted shift operator , Electron. J. Linear Algebra (2015), 944–963. ESOLVENT CRITERIA FOR SIMILARITY TO A NORMAL OPERATOR 22 [54] V. Vasyunin and S. Kupin,
Criteria for the similarity of a dissipative integral operatorto a normal operator , Algebra i Analiz (2001), no. 3, 65–104; English transl., St.Petersburg Math. J. (2002), no. 3, 389–416. School of Mathematics, Statistics and Physics, Herschel Building, Univer-sity of Newcastle, Newcastle upon Tyne, NE1 7RU, UK
E-mail address : [email protected] GMV Innovating Solutions, Tres Cantos 28760 (Madrid), Spain
E-mail address : [email protected] Departamento de Matem´aticas, Universidad Aut´onoma de Madrid, Canto-blanco 28049 (Madrid), Spain, and Instituto de Ciencias Matem´aticas (CSIC -UAM - UC3M - UCM)
E-mail address ::