aa r X i v : . [ m a t h . F A ] O c t SIMILARITY RESULTS FOR OPERATORS OF CLASS C RAPHA¨EL CLOU ˆATRE
Abstract. If T is a multiplicity-free contraction of class C with minimalfunction m T , then it is quasisimilar to the Jordan block S ( m T ). In case m T is a Blaschke product with simple roots forming a Carleson sequence, we showthat the relation between T and S ( m T ) can be strengthened to similarity.Under the additional assumption that u ( T ) has closed range for every innerdivisor u ∈ H ∞ of m T , the result also holds in the more general setting wherethe roots have bounded multiplicities. Introduction
Let T be a bounded linear operator on some Hilbert space. Assume that T is ofclass C and admits a cyclic vector. In this case, we know that T is quasisimilarto the Jordan block S ( θ ) where θ ∈ H ∞ is the minimal inner function of T (seeSection 2 for precise definitions). The problem we consider in this paper is asfollows: under the additional assumption that u ( T ) has closed range for everyinner divisor u ∈ H ∞ of θ , can the relation of quasisimilarity between T and S ( θ )be strenghtened to similarity ? We will motivate this question in the next section,but let us first make a few remarks. First, it is a classical fact that the converse ofthis statement holds, in fact it is known that u ( S ( θ )) is a partial isometry for everyinner divisor u of θ ; see [1]. Moreover, the special case of a diagonal operator givesan interpretation of Carleson’s interpolation theorem in our context of minimalfunctions of operators of class C . Our main result shows that this problem has apositive answer in the case where θ ( z ) = z m Y j ∈ N (cid:18) λ j | λ j | λ j − z − λ j z (cid:19) m j with sup j m j < ∞ and { λ j } j satisfying the Carleson condition(1) inf k ∈ N Y j = k (cid:12)(cid:12)(cid:12)(cid:12) λ j − λ k − λ j λ k (cid:12)(cid:12)(cid:12)(cid:12) > . The paper is organized as follows. Section 2 deals with preliminaries. Section3 treats the problem in the case where each m j is equal to one. The solutionhinges on the classical Carleson interpolation theorem. In Section 4, we give acharacterization of Carleson sequences { λ j } j ⊂ D in terms of operators of class C with minimal function Y j ∈ N (cid:18) λ j | λ j | λ j − z − λ j z (cid:19) . Mathematics Subject Classification.
Primary 47A45; Secondary 30E05.Research supported by NSERC (Canada) .
This motivates the assumption on the closure of the ranges. We then extend thesimilarity result to the case where the multiplicities of the roots are bounded. Inaddition to the statement and proof of the main result, Section 5 contains anothercrucial ingredient: an estimate concerning the norm of the similarity matrix betweennilpotent contractions and Jordan cells.It is appropriate here to address an insightful remark communicated to us bythe referee. Our main result Theorem 5.7 is concerned with a Carleson sequence { λ j } j ⊂ D where each λ j can have multiplicity at most M . This is a degenerate caseof the more general situation of a finite union of Carleson sequences, namely { λ j } j = S Mk =1 Λ k ⊂ D where Λ k ⊂ D is a Carleson sequence (with each of its elements havingmultiplicity one). It is a natural question whether Theorem 5.7 holds in this context.Using results from [5], [6], [14] and [10], it is possible to generalize Proposition5.2 after some significant modification of our present argument. However, a fullgeneralization of Theorem 5.7 would require an extension of Proposition 5.6 fromthe nilpotent case to that where the operator is merely algebraic. At this time, wefeel this is the major obstacle to obtaining the stronger theorem. It is our hopethat we may settle the question in future work.The author wishes to thank his advisor Hari Bercovici for his generosity, hismathematical insight and for suggesting the problem investigated in this paper.Moreover, the author is indebted to the referee for his generous and numerouscomments that helped improve the quality of this work.2. Background and motivation
Let H be a Hilbert space and T a bounded linear operator on H , which weindicate by T ∈ B ( H ). Assume that T is a completely non-unitary contraction.Let H ∞ be the Hardy space of bounded holomorphic functions on the unit disc D . The Sz.-Nagy–Foias H ∞ functional calculus then provides a contractive algebrahomomorphim Φ : H ∞ → B ( H ) such that Φ( p ) = p ( T ) for every polynomial p .Moreover, Φ is continuous when H ∞ and B ( H ) are given their respective weak-startopologies. We will write Φ( u ) = u ( T ) for u ∈ H ∞ . The contraction T is said tobe of class C if Φ has non-trivial kernel. It is known that ker Φ = m T H ∞ forsome inner function m T called the minimal function of T . The minimal functionis uniquely determined up to a scalar of absolute value one.For any inner function θ ∈ H ∞ , the space H ( θ ) = H ⊖ θH is closed andinvariant for S ∗ , the adjoint of the shift operator S on H . The operator S ( θ )defined by S ( θ ) ∗ = S ∗ | ( H ⊖ θH ) is called a Jordan block ; it is of class C withminimal function θ .A vector x ∈ H is said to be cyclic for T ∈ B ( H ) if the linear manifold generatedby { T n x : n ≥ } is dense in H . If an operator has a cyclic vector, it is called multiplicity-free . A bounded linear operator X : H → H ′ is called a quasiaffinity if it is injective and has dense range. The following is Theorem 3.2.3 of [1], itsconclusion is summarized by saying that T is quasisimilar to S ( m T ). Theorem 2.1.
Let T ∈ B ( H ) be a multiplicity-free operator of class C . Then,there exist quasiaffinities X : H → H ( m T ) and Y : H ( m T ) → H with the propertythat XT = S ( m T ) X and T Y = Y S ( m T ) . More details about all of the above background material can be found in [1].
IMILARITY RESULTS FOR OPERATORS OF CLASS C Throughout the paper, we use the following notation for Blaschke factors: for λ ∈ D , λ = 0 we set b λ ( z ) = λ | λ | λ − z − λz , and we also set b ( z ) = z .We now give motivation for the main problem that we address. Let { λ j } j ⊂ D be a Carleson sequence, that is a sequence satisfying (1). Set b ( z ) = Q j ∈ N b λ j ( z ) . The classical Carleson interpolation theorem (see [2], [4]) implies that the map H ∞ /bH ∞ → ℓ ∞ u + bH ∞
7→ { u ( λ j ) } j is a bounded algebra isomorphism. On the other hand, a consequence of the com-mutant lifting theorem (see [12], [7], [8]) is that the map H ∞ /bH ∞ → { S ( b ) } ′ u + bH ∞ u ( S ( b ))is an isometric algebra isomorphism. Thus, we see that there exists a boundedalgebra isomorphism between { S ( b ) } ′ and ℓ ∞ . Consider now the operator T = M j ∈ N S (cid:0) b λ j (cid:1) . It is obvious that T is of class C with minimal function b . Moreover, any vector L j ∈ N h j ∈ L j ∈ N H ( b λ j ) with h j = 0 for every j ∈ N is cyclic for T . Therefore, T is quasisimilar to S ( b ). If we denote by Id X the identity operator on a space X ,then we have S ( b λ j ) = λ j Id C , and thus { T } ′ = M j ∈ N a j : { a j } j ∈ ℓ ∞ ∼ = ℓ ∞ , so in fact { S ( b ) } ′ and { T } ′ are boundedly isomorphic as algebras. We thus see that T and S ( b ) share properties beyond what is guaranteed by mere quasisimilarity.We set out to investigate this phenomenon more carefully.3. Blaschke product with multiplicity one
Let { λ j } j ⊂ D satisfy the Carleson condition (1), and set b ( z ) = Y j ∈ N b λ j ( z ) . By the classical Carleson interpolation theorem, for every subset A ⊂ N , we canfind φ A ∈ H ∞ such that φ A ( λ j ) = ( j ∈ A j / ∈ A with k φ A k ≤ C ,where C is the so-called constant of interpolation (see [4] p.276)and is independent of the set A . RAPHA¨EL CLOUˆATRE
Lemma 3.1.
Let
Ψ : H ∞ → B ( H ) be a bounded algebra homomorphism such that bH ∞ ⊂ ker Ψ . For every A ⊂ N , define g A = Ψ( φ A ) − (Id H − Ψ( φ A )) ∈ B ( H ) . Then G = { g A : A ⊂ N } ⊂ B ( H ) is an abelian group under multiplication.Proof. We first need to check that g A is well-defined since the function φ A is notuniquely determined. Assume that φ and φ are two candidates for φ A . Then, φ − φ vanishes at every λ j , whence φ − φ = bf for some f ∈ H ∞ . Consequently,Ψ( φ − φ ) = Ψ( bf ) = 0 since Ψ vanishes on bH ∞ , and thus g A is well-defined. Astraightforward calculation now yields that Ψ( φ A ) is idempotent and that g A is aninvertible operator. Moreover, g A g B = g ( A ∩ B ) ∪ ( A c ∩ B c ) . (cid:3) Proposition 3.2.
Let T ∈ B ( H ) . Assume that there exists a bounded algebrahomomorphism Ψ : H ∞ → B ( H ) such that Ψ( p ) = p ( T ) for every polynomial p ,which is also continuous when H ∞ is given the weak-star topology and B ( H ) theweak operator topology . Furthermore, assume that H = W j ∈ N H j where H j =ker( T − λ j ) and { λ j } j ⊂ D is a Carleson sequence. Then there exists an invertibleoperator X ∈ B ( H ) such that the subspaces { XH j } j ∈ N are mutually othogonal and XT X − = L j ∈ N λ j Id XH j .Proof. Since H = W j ∈ N H j , it is easy to check that bH ∞ ⊂ ker Ψ. Moreover, thefact that Ψ is bounded implies k g A k ≤ k Ψ kk φ A k + 1for every A ⊂ N , and using the fact that k φ A k H ∞ ≤ C , we see that the inclusion ι : G → B ( H ) is strongly continuous and uniformly bounded. By Dixmier’s theorem(see [3] or Theorem 9.3 of [11]), there exists a bounded invertible operator X ∈ B ( H )such that Xg A X − ∈ B ( H ) is unitary for every A ⊂ N . Now, the equations( Xg A X − ) ∗ ( Xg A X − ) = Id H = ( Xg A X − )( Xg A X − ) ∗ along with the definition of g A easily yield that X Ψ( φ A ) X − must be normal.Hence, X Ψ( φ A ) X − is a self-adjoint projection. Notice that for j = k , φ { j } φ { k } = 0so that P j ∈ N X Ψ( φ { j } ) X − converges strongly to an element in B ( H ).For each j ∈ N , choose a sequence { p j,n } n of polynomials such that p j,n → φ { j } in the weak-star topology as n → ∞ . By the continuity property of Ψ, we haveΨ( p j,n ) → Ψ( φ { j } )in the weak operator topology as n → ∞ . In particular, we see that for x ∈ H k and y ∈ H , we have h Ψ( φ { j } ) x, y i = lim n →∞ h Ψ( p j,n ) x, y i = lim n →∞ h ( p j,n ( T ) | H k ) x, y i = lim n →∞ p j,n ( λ k ) h x, y i = φ { j } ( λ k ) h x, y i whence Ψ( φ { j } ) x = φ { j } ( λ k ) x for every x ∈ H k and every k ∈ N . By choice of φ { j } , this shows that Ψ( φ { j } ) = Id on H j , Ψ( φ { j } ) = 0 on H k for k = j and T x = λ k x = X j ∈ N λ j Ψ( φ { j } ) x IMILARITY RESULTS FOR OPERATORS OF CLASS C for every x ∈ H k and every k ∈ N . Since H = W k ∈ N H k , we get that XT X − = X j ∈ N λ j X Ψ( φ { j } ) X − . Recall now that every X Ψ( φ { j } ) X − is a self-adjoint projection. We conclude that XT X − is normal, so that XH j ⊥ XH k for j = k and XT X − = L j ∈ N λ j Id XH j . (cid:3) Corollary 3.3.
Let { λ j } j ⊂ D be a Carleson sequence and let T ∈ B ( H ) be anoperator of class C with minimal function b ( z ) = Q j ∈ N b λ j ( z ) . Then T is similarto L j ∈ N λ j Id H j where H j = ker( T − λ j ) for every j ∈ N .Proof. Notice that b is the least common inner multiple of the family { b λ j } j . ByTheorem 2.4.6 in [1], we can thus write H = ker b ( T ) = _ j ∈ N ker b λ j ( T ) = _ j ∈ N H j . Apply now Proposition 3.2 to T with Ψ being the usual Sz.-Nagy–Foias H ∞ func-tional calculus. We see that there exists an invertible operator X ∈ B ( H ) such thatthe subspaces { XH j } j ∈ N are mutually othogonal and XT X − = L j ∈ N λ j Id XH j .Let H ′ = L j ∈ N H j be the external orthogonal sum of the subspaces H j ⊂ H ,and let ι j : H j → H ′ denote the canonical inclusion. Define X ′ : H ′ → H as X ′ | ι j ( H j ) = X | H j . Using that XH j ⊥ XH k for j = k , it is easy to verify that X ′ is bounded and invertible, and that it establishes a similarity between L j ∈ N λ j Id H j and L j ∈ N λ j Id XH j . The proof is complete. (cid:3) Characterization of Carleson sequences
We now want to give a characterization of the sequences { λ j } j ⊂ D which satisfythe Carleson condition (1). This is done in Theorem 4.4. Assume that { λ j } j ⊂ D is a Blaschke sequence of distinct points. Let b ( z ) = Y j ∈ N b λ j ( z ) . Lemma 4.1.
Let D = L j ∈ N S ( b λ j ) and u ∈ H ∞ be an inner divisor of b . Then u ( D ) has closed range if and only if inf {| u ( λ j ) | : u ( λ j ) = 0 } > . Proof.
It is easy to verify that D = L j ∈ N λ j Id C . Let F = { j ∈ N : u ( λ j ) = 0 } .Then u ( D ) = L j ∈ N u ( λ j ) has closed range if and only if L j ∈ F u ( λ j ) has closedrange. But L j ∈ F u ( λ j ) is clearly injective and has dense range, so that L j ∈ F u ( λ j )has closed range if and only L j ∈ F u ( λ j ) is invertible, which in turn is equivalentto the condition of the lemma. (cid:3) For n ∈ N , put u n ( z ) = b ( z ) b λ n ( z ) = Y j = n (cid:18) λ j | λ j | λ j − z − λ j z (cid:19) . Moreover, we will need a notation for partial products of an infinite product. Given f ( z ) = Q j f j ( z ), denote by f ( z ; m ) the m -th partial product Q mj =1 f j ( z ). RAPHA¨EL CLOUˆATRE
Lemma 4.2.
Assume that u n ( λ n ) → as n → ∞ . Then there exists an innerdivisor u of b such that inf {| u ( λ j ) | : u ( λ j ) = 0 } = 0 . Proof.
Define α k = 1+2 − ( k − . Choose n ∈ N with the property that | u n ( λ n ) | < − . Choose m > n such that | u n ( λ n ; m ) | < − and note that u n ( λ n ; m ) =0. Set w ( z ) = u n ( z ; m ) . Next, choose n > m with the property that | u n ( λ n ) | < − . Note that | b λ ( z ) | → | λ | → z ∈ D , so we can also require that | b λ n ( λ n ) | ≥ α − . Define v ( z ) = Y j = n ,n (cid:18) λ j | λ j | λ j − z − λ j z (cid:19) . Notice that v b λ n = u n and v b λ n = u n . Hence, | v ( λ n ) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) u n ( λ n ) b λ n ( λ n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ α and | v ( λ n ) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) u n ( λ n ) b λ n ( λ n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ α where we used the fact that | b x ( y ) | = | b y ( x ) | . Choose m > n such that | v ( λ n ; m ) | ≤ α and | v ( λ n ; m ) | ≤ α . Set w ( z ) = v ( z ; m ) . Assume that we have chosen n , . . . , n k − , m , . . . , m k − ∈ N with m j > n j >m j − > n j − for 2 ≤ j ≤ k − v , . . . , v k − , w , . . . , w k − as above with theproperty that | v j ( λ j ) | ≤ j +1 j Y µ =2 α µ and | v j ( λ p ) | ≤ p +1 j Y µ =2 α µ for every 1 ≤ p < j ≤ k −
1. We will construct w k . Choose n k > m k − ∈ N suchthat | u n k ( λ n k ) | < − ( k +1) and | b λ nk ( λ n j ) | ≥ α k for every j = 1 , . . . , k −
1. Define v k ( z ) = Y j = n ,...,n k (cid:18) λ j | λ j | λ j − z − λ j z (cid:19) . IMILARITY RESULTS FOR OPERATORS OF CLASS C Notice that v k b λ nk = v k − and v k Q k − j =1 b λ nj = u n k . Hence, | v k ( λ n k ) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) u n k ( λ n k ) Q k − µ =1 b λ nµ ( λ n k ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ α k − k k +1 ≤ k +1 k Y µ =2 α µ and | v k ( λ n j ) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) v k − ( λ n j ) b λ nk ( λ n j ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ j +1 k Y µ =2 α µ for j = 1 , . . . , k −
1. Choose m k ∈ N such that | v k ( λ n k ; m k ) | ≤ k k Y µ =2 α µ and | v k ( λ n j ; m k ) | ≤ j k Y µ =2 α µ for j = 1 , . . . , k −
1. Set w k ( z ) = v k ( z ; m k ) . Finally, define u = lim k w k . It is straightforward to check that for every p ∈ N the function u defines an inner divisor of b/b λ np , whence u ( λ n p ) = 0 and u is aninner divisor of b . It remains to show that u ( λ n p ) → p → ∞ . By definition,we have | u ( λ n p ) | = lim k | w k ( λ n p ) | ≤ p lim k k Y µ =2 α µ , so that | u ( λ n p ) | → p → ∞ since Q µ α µ is convergent. (cid:3) Proposition 4.3.
Let { λ j } j ⊂ D be a Blaschke sequence of distinct points. Let D = L j ∈ N S ( b λ j ) . Then { λ j } j ⊂ D is a Carelson sequence if and only if u ( D ) hasclosed range for every inner divisor u ∈ H ∞ of b .Proof. Assume that { λ j } j is not a Carleson sequence. This means thatinf k ∈ N | u k ( λ k ) | = inf k ∈ N Y j = k (cid:12)(cid:12)(cid:12)(cid:12) λ j − λ k − λ j λ k (cid:12)(cid:12)(cid:12)(cid:12) = 0so we can find a sequence { u k n } n such that u k n ( λ k n ) → n → ∞ . By Lemma4.1 and 4.2, we see that there exists some inner divisor u of b for which u ( D ) doesn’thave closed range.Conversely, assume that { λ j } j is a Carleson sequence, and let u ∈ H ∞ be aninner divisor of b . Assume that u ( λ j ) = 0. Since u divides b , by definition of u j wesee that there must exist an inner function ψ ∈ H ∞ such that u j = ψu . In turn,this implies that | u ( λ j ) | ≥ | u j ( λ j ) | . By assumption, we know that inf j | u j ( λ j ) | > {| u ( λ j ) | : u ( λ j ) = 0 } > . By virtue of Lemma 4.1, this completes the proof. (cid:3)
We can now prove the main result of this section, which gives another interpre-tation of the classical Carleson interpolation theorem.
RAPHA¨EL CLOUˆATRE
Theorem 4.4.
Let { λ j } j ⊂ D be a Blaschke sequence of distinct points and b = Q j ∈ N b λ j be the corresponding Blaschke product. The following statementsare equivalent:(i) { λ j } j ⊂ D is a Carleson sequence(ii) every operator T of class C with minimal function b is similar to L j ∈ N λ j Id H j where H j = ker( T − λ j ) for every j ∈ N (iii) every multiplicity-free operator T of class C with minimal function b is sim-ilar to S ( b ) (iv) u ( T ) has closed range for every multiplicity-free operator T of class C withminimal function b and every inner divisor u ∈ H ∞ of b (v) u ( D ) has closed range for every inner divisor u ∈ H ∞ of b , where D = L j ∈ N S ( b λ j ) .Proof. Corollary 3.3 shows that (i) implies (ii).Assume that (ii) holds and let T be a multiplicity-free operator of class C withminimal function b . Set H j = ker( T − λ j ) and K j = ker( S ( b ) − λ j ) for every j ∈ N ,and choose X : H → H ( b ) and Y : H ( b ) → H quasiaffinities such that XT = S ( b ) X and T Y = Y S ( b ). A routine verification shows that K j has dimension 1. Now, XH j = X ker( T − λ j ) ⊂ ker( S ( b ) − λ j ) = K j and Y K j = Y ker( S ( b ) − λ j ) ⊂ ker( T − λ j ) = H j . Since X and Y are injective, we find that H j also has dimension1 and XH j = K j . Corollary 3.3 then implies that both T and S ( b ) are similar to L j ∈ N λ j Id C , which in turn implies (iii).Assume that (iii) holds. Then φ ( T ) is similar to φ ( S ( b )) for every φ ∈ H ∞ .Now, it is a classical fact that u ( S ( b )) is a partial isometry for every inner divisor u ∈ H ∞ of b (see problem 11 p.43 of [1]), so that u ( T ) has closed range for everyinner divisor of u ∈ H ∞ b , which is (iv).Assume that (iv) holds. As was noted in Section 2, D is a multiplicity-freeoperator of class C with minimal function b . Assertion (v) obviously follows.Finally, the fact that (v) implies (i) follows from Proposition 4.3. (cid:3) Blaschke product with bounded multiplicity
We are now ready to address the main question in the case where the multiplic-ities of the roots of the minimal function are bounded. As before, let { λ j } j ⊂ D bea Blaschke sequence of distinct points. Throughout this section we assume that thesequence { λ j } j satisfies the Carleson condition (1). Let { m j } j ⊂ N be a boundedsequence, set M = sup j m j and define θ ( z ) = ∞ Y j =1 b m j λ j ( z ) . By the interpolation theorem for germs of holomorphic functions (from [13],[14], seealso Chapter 9 Section 4 of [10]), for every u ∈ H ∞ there exists a function b u ∈ H ∞ with the property that b u ( λ j ) = u ( λ j )and d p b udz p ( λ j ) = 0for every j ∈ N , 1 ≤ p ≤ M . Moreover, we have that k b u k H ∞ ≤ C k{ u ( λ j ) } j k ℓ ∞ ≤ C k u k H ∞ IMILARITY RESULTS FOR OPERATORS OF CLASS C for some constant C > u (this follows from the open mappingtheorem). Denote by χ the identity function on D . Lemma 5.1.
Let T ∈ B ( H ) be an operator of class C with minimal function θ .Then the map Ψ : H ∞ → B ( H ) u b u ( T ) is a bounded algebra homomorphism such that Ψ( p ) = p ( b χ ( T )) for every polynomial p . Moreover, Ψ is continuous when H ∞ is given the weak-star topology and B ( H ) the weak operator topology. Finally, ker( b χ ( T ) − λ j ) = ker b m j λ j ( T ) for every j ∈ N and H = W j ∈ N ker b m j λ j ( T ) .Proof. Arguing as in the proof of Corollary 3.3, we see that H = _ j ∈ N ker b m j λ j ( T ) . Set H j = ker b m j λ j ( T ) . A routine verification shows that u ( T ) | H j = u ( T | H j ) forevery j ∈ N and u ∈ H ∞ . Notice now that b u − u ( λ j ) has a zero of order M at λ j ,so that b m j λ j divides b u − u ( λ j ), whence b u ( T ) = u ( λ j ) on H j . In particular, we see that H j ⊂ ker( b χ ( T ) − λ j ). Choose x ∈ ker( b χ ( T ) − λ j ). The restriction of T to W ∞ n =0 T n x is still of class C (see [1]) and hence it has a minimal function, say m x . Then m x divides the greatest common inner divisor of θ and b χ − λ j , and thus m x divides b m j λ j , which implies that x ∈ H j . In other words, H j = ker b m j λ j ( T ) = ker( b χ ( T ) − λ j ).Let us now show that Ψ is well-defined. Assume that c u and c u are two boundedholomorphic functions on D having the same interpolating property as b u . Then, c u − c u has a zero of order M at every λ j , and thus θ divides c u − c u . But θ is theminimal function of T , so that ( c u − c u )( T ) = 0 and Ψ is well-defined.It is easily checked that Ψ is bounded: k Ψ( u ) k = k b u ( T ) k ≤ k b u k ≤ C k u k . Choose now a sequence { u n } n ⊂ H ∞ such that u n → u in the weak-star topology,i.e. u n ( λ ) → u ( λ ) for every λ ∈ D , and sup n k u n k < ∞ . Since Ψ is bounded and H = W j ∈ N H j , to check the desired continuity property it suffices to verify that h Ψ( u ) x, y i = lim n →∞ h Ψ( u n ) x, y i for any y ∈ H and x ∈ H j , j ∈ N . Let us then pick x ∈ H j , y ∈ H . We alreadyestablished that b v ( T ) = v ( λ j ) on H j for every v ∈ H ∞ , so we see that h Ψ( u ) x, y i = h b u ( T ) x, y i = h u ( λ j ) x, y i = lim n →∞ h u n ( λ j ) x, y i = lim n h c u n ( T ) x, y i = lim n →∞ h Ψ( u n ) x, y i and Ψ has the announced continuity property.Finally, if p is a polynomial and x ∈ H j , thenΨ( p ) x = b p ( T ) x = p ( λ j ) x = p ( b χ ( T ) | H j ) x = p ( b χ ( T )) x and, again, since H = W j ∈ N H j we conclude that Ψ( p ) = p ( b χ ( T )). The homomor-phism property is verified in a similar fashion. (cid:3) Proposition 5.2.
Let T ∈ B ( H ) be an operator of class C with minimal function θ ( z ) = ∞ Y j =1 b m j λ j ( z ) where { λ j } j ⊂ D is a Carleson sequence and sup j m j = M < ∞ . Then T is similarto L j ∈ N T | H j , where H j = ker b m j λ j ( T ) .Proof. Apply Proposition 3.2 to the operator b χ ( T ) (see Lemma 5.1) to get thatthere exists an invertible operator R ∈ B ( H ) such that RH j ⊥ RH k for j = k where H j = ker( b χ ( T ) − λ j ) = ker b m j λ j ( T ) for j ∈ N . We may let A j = RT R − | RH j to get that RT R − = L j ∈ N A j . Let H ′ = L j ∈ N H j be the external orthogonalsum of the subspaces H j ⊂ H , and let ι j : H j → H ′ denote the canonical inclusion.Define R ′ : H ′ → H as R ′ | ι j ( H j ) = R | H j . Using that RH j ⊥ RH k for j = k , it iseasy to verify that R ′ is bounded and invertible, and that it establishes a similaritybetween L j ∈ N A j and L j ∈ N T | H j . The proof is complete. (cid:3)
If in addition we assume that T is multiplicity-free, then we obtain that T | H j is similar to S ( b m j λ j ). Indeed, choose X : H → H ⊖ θH and Y : H ⊖ θH → H quasiaffinities such that XT = S ( θ ) X and T Y = Y S ( θ ). Arguing as in the proofof Theorem 4.4, it is easy to see that XH j = X ker b m j λ j ( T ) = ker b m j λ j ( S ( θ )). Inparticular, X | H j establishes a similarity between T | H j and S ( θ ) | ker b m j λ j ( S ( θ )),which in turn is unitarily equivalent to S ( b m j λ j ) by Proposition 3.1.10 in [1]. Thisestalishes the claim. However, this procedure doesn’t offer any control over thenorm of ( X | H j ) − , which turns out to be crucial in the proof of our main result.The following development remedies this matter.Before proceeding we need to recall some classical results from interpolation the-ory. The first one is a combination of Lemma 3.2.8, Corollary 3.2.11 and Theorem3.2.14 of [9]. It is originally from [14]. Proposition 5.3.
Let { θ j } j ∈ N be inner functions such that θ = Q j θ j is their leastcommon inner multiple. The following statements are equivalent:(i) there exists a constant c > independent of z with the property that | θ ( z ) | ≥ c inf j ∈ N | θ j ( z ) | for every z ∈ D (ii) there exists a constant c > with the property that for any finite set σ ⊂ N there are functions f σ , g σ ∈ H ∞ satisfying f σ θ σ + g σ θ N \ σ = 1 with k f σ k ≤ c , where θ σ = Q j ∈ σ θ j (iii) inf σ ⊂ N inf z ∈ D {| θ σ ( z ) | + | θ N \ σ ( z ) |} > . The following is Lemma 3.2.18 from [9].
Proposition 5.4.
A sequence { λ j } j ⊂ D is a Carleson sequence if and only ifstatement (1) above holds with θ j = b λ j . We can now establish an important technical result.
Lemma 5.5.
Let { λ j } j ⊂ D be a Carleson sequence and T = L j ∈ N T j be anoperator of class C acting on H = L j ∈ N H j , with minimal function θ and suchthat b m j λ j ( T j ) = 0 . Assume also that u ( T ) has closed range for every inner divisor IMILARITY RESULTS FOR OPERATORS OF CLASS C u ∈ H ∞ of θ . Then the operator L j ∈ N b k j j ( T j ) has closed range for any choice ofinteger sequence { k j } j satisfying ≤ k j ≤ m j for every j ∈ N .Proof. By Proposition 5.4, we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Y j ∈ N b λ j ( z ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ c inf j ∈ N | b λ j ( z ) | for some constant c > z ∈ D . But then Proposition 5.3 impliesinf σ ⊂ N inf z ∈ D (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Y j ∈ σ b λ j ( z ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Y j / ∈ σ b λ j ( z ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) > . Hence, inf σ ⊂ N inf z ∈ D (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Y j ∈ σ b Mλ j ( z ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Y j / ∈ σ b Mλ j ( z ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) > m j ≤ M and b λ j is inner for every j ∈ N , we may writeinf σ ⊂ N inf z ∈ D (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Y j ∈ σ b m j λ j ( z ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Y j / ∈ σ b m j λ j ( z ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ inf σ ⊂ N inf z ∈ D (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Y j ∈ σ b Mλ j ( z ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Y j / ∈ σ b Mλ j ( z ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) > . Applying Proposition 5.3 again, we find that there exists a constant c > j ∈ N there exists functions f j , g j ∈ H ∞ satisfying f j θb m j λ j + g j b m j λ j = 1and k f j k ≤ c . In particular, we haveId H j = f j ( T j ) θb m j λ j ( T j ) + g j ( T j ) b m j λ j ( T j ) = f j ( T j ) θb m j λ j ( T j ) = θb m j λ j ( T j ) f j ( T j )or f j ( T j ) = θb m j λ j ( T j ) ! − . Note also that k f j ( T j ) k ≤ c . Now, let φ = Q r ∈ N b k r λ r . Since 0 ≤ k j ≤ m j , we canwrite θb m j λ j = φb k j λ j ψ j for some inner function ψ j ∈ H ∞ and we get the relation θb m j λ j ( T j ) = φb k j λ j ( T j ) ψ j ( T j ) = ψ j ( T j ) φb k j λ j ( T j . Since θ/ ( b m j λ j )( T j ) is invertible, we conclude that φ/ ( b k j λ j )( T j ) is actually invertible.We have φb k j λ j ( T j ) − = ψ j ( T j ) f j ( T j )and we infer (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) φb k j λ j ( T j ) − (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ c k ψ j ( T j ) k ≤ c since ψ j is inner for every j ∈ N . This shows that the operator M j ∈ N φb k j λ j ( T j )is invertible, and we may write M j ∈ N b k j λ j ( T j ) = M j ∈ N b k j λ j ( T j ) φb k j λ j ( T j ) φb k j λ j ( T j ) − = M j ∈ N b k j λ j ( T j ) φb k j λ j ( T j ) M j ∈ N φb k j λ j ( T j ) − = M j ∈ N φ ( T j ) M j ∈ N φb k j λ j ( T j ) − = φ ( T ) M j ∈ N φb k j λ j ( T j ) − which completes the proof since φ is an inner divisor of θ , and consequently φ ( T )has closed range by assumption. (cid:3) The last ingredient we need is the following. For m ∈ N , denote by χ m thefunction z z m on D . Proposition 5.6.
Let H be an n -dimensional Hilbert space. Let N ∈ B ( H ) bea multiplicity-free contraction which is nilpotent of order n . Then there exists aninvertible operator X such that XN X − = S ( χ n ) with the additional property that k X k ≤ and k X − k ≤ ε − n − , where ε > satisfies k N x k ≥ ε k x k for every x ∈ (ker N ) ⊥ .Proof. There is no loss of generality in assuming that H = C n . Let ξ ∈ C n be acyclic vector for N . We may additionnally assume that k N n − ξ k = 1. Define now ζ n − = N n − ξ,ζ n − = N n − ξ − a n − N n − ξ,ζ n − = N n − ξ − a n − N n − ξ − a n − N n − ξ, · · · ζ = ξ − a n − N ξ − . . . − a N n − ξ where a , . . . , a n − ∈ C are chosen such that h ζ n − , ζ k i = 0 for every 0 ≤ k ≤ n − ζ is a cyclic vector for N , and hence we may assume that N n − ξ isorthogonal to N k ξ for every 0 ≤ k ≤ n − IMILARITY RESULTS FOR OPERATORS OF CLASS C Denote by e , . . . , e n − the usual orthonormal basis of C n . Since ξ is cyclic, thevectors ξ, N ξ, . . . , N n − ξ form another (possibly non-orthonormal) basis for C n .Define X : C n → C n as XN k ξ = e k . It is clear then that X is an invertiblelinear operator, and we have that XN X − e n − = XN n ξ = 0 and XN X − e k = XN k +1 ξ = e k +1 for 0 ≤ k ≤ n −
2. In other words,
XN X − = S ( χ n ).Let us now examine the norm of X and X − . Let c , . . . , c n − be arbitrarycomplex numbers. Using that N n − ξ ⊥ N k ξ for every 0 ≤ k ≤ n − k N n − ξ k = 1, we see that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n − X j =0 c j N j ξ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n − X j =0 c j N j ξ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) + | c n − | . Now, k N k ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n − X j =0 c j N j ξ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≥ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n − X j =0 c j N j +1 ξ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) + | c n − | = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n − X j =0 c j N j +1 ξ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) + | c n − | + | c n − | . Iterating this process yields (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n − X j =0 c j N j ξ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≥ n − X j =0 | c j | = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n − X j =0 c j e j (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) . In other words, k X k ≤
1. On the other hand, notice that ker N = C N n − ξ so that(ker N ) ⊥ = n − _ j =0 C N j ξ. In particular, we see that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n − X j =0 c j N j ξ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ ε (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n − X j =0 c j N j +1 ξ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) which implies (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n − X j =0 c j N j ξ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n − X j =0 c j N j ξ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) + | c n − | ≤ ε (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n − X j =0 c j N j +1 ξ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) + | c n − | = 1 ε (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n − X j =0 c j N j +1 ξ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) + 1 ε | c n − | + | c n − | . Iterating this calculation yields (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n − X j =0 c j N j ξ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ n − X j =0 | c j | ε n − − j ) ≤ ε n − n − X j =0 | c j | = 1 ε n − (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) n − X j =0 c j e j (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) . Thus, k X − k ≤ ε − n − . (cid:3) We are now ready to prove the main result of the paper.
Theorem 5.7.
Assume that { λ j } j ⊂ D is a Carleson sequence sequence and { m j } j ⊂ N is a bounded sequence. Let T ∈ B ( H ) be a multiplicity-free oper-ator of class C with minimal function θ ( z ) = Q j ∈ N b m j λ j ( z ) . Assume also that u ( T ) has closed range for every inner divisor u ∈ H ∞ of θ . Then T is similar to L j ∈ N S ( b m j λ j ) .Proof. By Proposition 5.2, we may assume that T = L j ∈ N T | H j where H j =ker b m j λ j ( T ). Suppose for the moment that for each j ∈ N there exists an invertibleoperator X j such that X j b λ j ( T | H j ) X − j = S ( χ m j ) andsup j ∈ N {k X j k , k X − j k} < ∞ . It is known that b λ j ( S ( b m j λ j )) is unitarily equivalent to S ( χ m j ) (see problem 2 p.42of [1]), so we may assume that X j b λ j ( T | H j ) X − j = b λ j ( S ( b m j λ j )). Now, we have b λ j ◦ b λ j = χ on D , and thus X j ( T | H j ) X − j = S ( b m j λ j ). The bounded operator X = L j ∈ N X j is invertible and provides a similarity between L j ∈ N T | H j and L j ∈ N S ( b m j λ j ), and the theorem follows.Hence we are left with the task of finding such invertible operators X j for j ∈ N .By Lemma 5.5, we have that L j ∈ N b λ j ( T | H j ) has closed range, so that there exists ε > j ) with the property that k b λ j ( T | H j ) x k ≥ ε k x k for every x ∈ (ker b λ j ( T | H j )) ⊥ and every j ∈ N . Note also that the operator b λ j ( S ( b m j λ j )) is multiplicity-free since it is unitarily equivalent to the Jordan block S ( χ m j ), as noted above. The same is true for b λ j ( T | H j ) by the remark followingProposition 5.2. Proposition 5.6 then immediately implies that for every j ∈ N we can find an invertible operator X j intertwining b j ( T | H j ) and S ( χ m j ) with theadditional property that sup j ∈ N {k X j k , k X − j k} < ∞ since sup j ∈ N m j < ∞ . (cid:3) Finally, we can answer our original question in the case where the multiplicities m j are uniformly bounded. Corollary 5.8.
Assume that { λ j } j ⊂ D is a Carleson sequence sequence and { m j } j ⊂ N is a bounded sequence. Let T ∈ B ( H ) be a multiplicity-free operator ofclass C with minimal function θ ( z ) = ∞ Y j =1 (cid:18) λ j λ j λ j − z − λ j z (cid:19) m j . IMILARITY RESULTS FOR OPERATORS OF CLASS C Assume that u ( T ) has closed range for every inner divisor u ∈ H ∞ of θ . Then T is similar to S ( θ ) .Proof. Both T and S ( θ ) satisfy the conditions of Theorem 5.7, and thus they aresimilar. (cid:3) References [1] Bercovici, H. Operator theory and arithmetic in H ∞ . Mathematical Surveys and Monographs,26. American Mathematical Society, Providence, RI, 1988[2] Carleson, L. An interpolation problem for bounded analytic functions. Amer. J. Math., 80(1958), 921-930[3] Dixmier, J. Les moyennes invariantes dans les semi-groupes et leurs applications. (French)Acta Sci. Math. Szeged 12 (1950), 213-227[4] Garnett, J. B. Bounded analytic functions. Pure and Applied Mathematics, 96. AcademicPress, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1981[5] Hartmann, A. Une approche de l’interpolation libre g´en´eralis´ee par la th´eorie des op´erateurset caract´erisation des traces H p | Λ. J. Operator Theory 35 (1996), no. 2, 281-316[6] Hartmann, A. Traces of certain classes of holomorphic functions on finite unions of Carlesonsequences. Glasg. Math. J. 41 (1999), no. 1, 103-114[7] Sz.-Nagy, B.; Foias, C. Dilatation des commutants d’op´erateurs. (French) C. R. Acad. Sci.Paris Sr. (1968),493-495[8] Sz.-Nagy, B.; Foias, C.; Bercovici, H.; Krchy, L. Harmonic analysis of operators on Hilbertspace. Second edition. Revised and enlarged edition. Universitext. Springer, New York, 2010[9] Nikolski, N. K. Operators, functions, and systems: an easy reading. Vol. 2. Model operatorsand systems. Translated from the French by Andreas Hartmann and revised by the author.Mathematical Surveys and Monographs, 93. American Mathematical Society, Providence, RI,2002[10] Nikolskii, N. K., Treatise on the shift operator. Spectral function theory. With an appendix byS. V. Hruc(ev [S. V. Khrushchv] and V. V. Peller. Translated from the Russian by Jaak Peetre.Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of MathematicalSciences], 273. Springer-Verlag, Berlin, 1986[11] Paulsen, V. Completely bounded maps and operator algebras. Cambridge Studies in Ad-vanced Mathematics, 78. Cambridge University Press, Cambridge, 2002[12] Sarason, D. Generalized interpolation in H ∞ . Trans. Amer. Math. Soc. 127, (1967), 179-203[13] Vasjunin, V. I. Unconditionally convergent spectral expansions, and nonclassical interpola-tion. (Russian) Dokl. Akad. Nauk SSSR 227 (1976), no. 1, 11-14[14] Vasjunin, V. I. Unconditionally convergent spectral decompositions and interpolation prob-lems. (Russian) Spectral theory of functions and operators. Trudy Mat. Inst. Steklov. 130(1978), 5-49, 223 Department of Mathematics, Indiana University, 831 East 3rd Street, Bloomington,IN 47405
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