A note on k -hyperreflexivity of Toeplitz-harmonic subspaces
aa r X i v : . [ m a t h . F A ] N ov A NOTE ON k -HYPERREFLEXIVITY OFTOEPLITZ-HARMONIC SUBSPACES P. BUDZY ´NSKI, K. PIWOWARCZYK AND M. PTAK
Abstract.
The 2-hyperreflexivity of Toeplitz-harmonic type subspace gener-ated by an isometry or a quasinormal operator is shown. The k -hyperreflexivityof the tensor product S ⊗ V of a k -hyperreflexive decomposable subspace S and an abelian von Neumann algebra V is established. Introduction
The concepts of reflexivity, transitivity and hyperreflexivity arise from the in-variant subspace problem. An algebra of operators is reflexive if it has so many(common) invariant subspaces that they determine the algebra itself or, equiva-lently, there are so many rank one operators in the preanihilator of the algebrathat they characterize the algebra. The latter condition enables generalization ofthe reflexivity concept to subspaces of operators. The transitivity, in contrast tothe reflexivity, means that there are no rank one operators in the preanihilator. Asubspace of operators is hyperreflexive if the standard (norm) distance from anyoperator to the subspace is controlled by the distance induced by rank one operators(equivalently, in case of an algebra, the distance induced by invariant subspaces).The concepts of k -reflexivity and k -hyperreflexivity are natural generalizations ofreflexivity and hyperreflexivity (rank one operators are replaced by rank k operatorsin relevant conditions). Certain subspaces, as being transitive, are far away frombeing reflexive. Nevertheless, they turn out to be 2-reflexive or 2-hyperreflexive.The space of all Toeplitz operators on the Hardy space on the unit disc is a primaryexample for this – it is transitive (cf. [1]) and 2-hyperreflexive (cf. [12]). The samephenomenon occurs in the case of the space of all Toeplitz operators on the Hardyspace on the polydisc (cf. [14]).The smallest weak ∗ closed subspace containing all powers of a given operator A and all powers of its adjoint is called a Toeplitz-harmonic subspace generated by A . Clearly, the space T ( D ) of all Toeplitz operators on the Hardy space H ( D ) is a Mathematics Subject Classification.
Primary: 47L05; Secondary: 47L75.
Key words and phrases. k -hyperreflexive subspace, direct integral, tensor product, isometry, quasi-normal operator.The research of the first author was partially supported by the NCN (National Science Center)grant DEC-2011/01/D/ST1/05805. Toeplitz-harmonic subspace generated by the operator T z of multiplication by theindependent variable. Hyperreflexivity of a Toeplitz-harmonic subspace generatedby a C contraction was shown in [4]. This result, in particular, implies hyper-reflexivity of the Toeplitz-harmonic subspace generated by the Bergman operator T z – the operator of multiplication by independent variable acting on the Bergmanspace on the unit disc. On the other hand, the Toeplitz-harmonic subspace gener-ated by T z acting on H ( D ) is transitive, thus it is not hyperreflexive. In view ofthis, it seems natural to ask about k -hyperreflexivity ( k >
2) of a Toeplitz-harmonicsubspace generated by an isometry or a quasinormal operator. In this paper weprove that such a subspace is 2-hyperreflexive (cf. Theorems 8 and 9). The proofis led via direct integral theory. For this purpose we prove that tensor product of a k -hyperreflexive decomposable subspace which has property A /k (1) and an abelianvon Neumann algebra is k -hyperreflexive (cf. Proposition 5).2. Preliminaries
In all what follows, by Z and by N we denote the set of all integers and allnon-negative integers, respectively. Denote also by ˆ N the set N ∪ {∞} . If X is alinear space and Y is a subset of X , then lin Y stands for the linear span of Y .Suppose that H is a (complex and separable) Hilbert space. Let B ( H ) denotethe Banach algebra of all bounded linear operators on H . For k ∈ N , F k ( H ) standsfor the set of operators on H of rank at most k . The set of trace class operators on H will be denoted by T ( H ). If T ∈ B ( H ), then W ( T ) stands for the WOT closedalgebra generated by T and the identity operator I . For a family S of operators in B ( H ), w ∗ -cl S denotes the weak ∗ closure of S .Suppose that S ⊆ B ( H ) is a (linear) subspace. For an operator A ∈ B ( H ) and k ∈ N we consider the following quantities d ( A, S ) = inf {k A − T k : T ∈ S} , α k ( A, S ) = sup {|h A, t i| : t ∈ S ⊥ ∩ B k ( H ) } , where h A, t i = tr( At ), S ⊥ = { t ∈ T ( H ) : h T, t i = 0 for all T ∈ S} and B k ( H )stands for the unit ball in F k ( H ) (with respect to trace norm k · k ). Recall that d ( A, S ) > α k ( A, S ) for every A ∈ B ( H ). The subspace S is called k -hyperreflexive if there is a constant C such that d ( A, S ) C α k ( A, S ) , A ∈ B ( H ) . (1)By κ k ( S ) we denote the infimum of the collection of all constants C such that(1) holds. An operator T ∈ B ( H ) is said to be k -hyperreflexive if W ( T ) is k -hyperreflexive. For more background of reflexivity and k -hyperreflexivity see [3]and [12]. NOTE ON k -HYPERREFLEXIVITY OF TOEPLITZ-HARMONIC SUBSPACES 3 It is known that k -hyperreflexivity is not hereditary in general, i.e., a subspaceof k -hyperreflexive subspace need not to be k -hyperreflexive itself. The situationimproves if the subspace has property A /k ( r ) (cf. [12, Proposition 3.8]). Let k ∈ N and r ∈ [1 , ∞ ). Recall that a weak ∗ closed subspace S of B ( H ) has property A /k ( r )if for every weak ∗ continuous functional ψ : B ( H ) → C and every ε > f ∈ F k ( H ) such that k f k ( r + ε ) k ψ k and ψ ( T ) = h T, f i for all T ∈ S . Directintegral of subspaces which have property A /k (1) has this property as well (cf. [7,Theorem 3.6] and Lemma 2). We will frequently use the following fact, which is ageneralization of a result due to Kraus and Larson in [13, Theorem 3.3]. Lemma 1. [12, Proposition 3.8]
Let k ∈ N and r ∈ [1 , ∞ ) . If S is a k -hyper-reflexive subspace of B ( H ) which has property A /k ( r ) , then any weak ∗ closed sub-space S of S is k -hyperreflexive and κ k ( S ) r + ( r + 1) κ k ( S ) . Let us now recall some basic definitions concerning direct integrals of subspaces(we refer the reader to monographs [6] and [17] for more information on the directintegral theory). Let (cid:0) Λ, B , µ (cid:1) be a measure space, where Λ is a separable metricspace, B is the σ -algebra of all Borel subsets of Λ and µ is a σ -finite regular Borelmeasure on Λ . Let H ⊆ H ⊆ . . . ⊆ H ∞ be a sequence of Hilbert spaces. Supposethat { Λ n : n ∈ ˆ N } ⊆ B is a partition of Λ . Let H ( λ ) = H n for λ ∈ Λ n and n ∈ ˆ N . Then H = R ⊕ Λ H ( λ ) µ (d λ ) denotes the Hilbert space of all (equivalenceclasses of) B -measurable H ∞ -valued functions on Λ such that for µ -a.e. λ ∈ Λ , f ( λ ) ∈ H ( λ ) and R Λ k f ( λ ) k µ (d λ ) < ∞ . Throughout the rest of the paper, D ( H )(resp. D ′ ( H )) will stand for the set of all diagonal (resp. decomposable) operatorsin B ( H ) ( D ′ ( H ) is indeed a commutant of D ( H ); cf. [17, Lemma I.3.2]). Supposethat S is an weak ∗ closed subspace of D ′ ( H ) such that there exists a countablegenerating set { T n : n ∈ N } for S . Such a subspace S is said to be decomposable .For λ ∈ Λ , let S ( λ ) be the weak ∗ closed subspace generated by { T n ( λ ) : n ∈ N } . Itis a matter of verification that definition of S ( λ ) does not depend on the choice of agenerating set (cf. [10, p. 1397]). The family {S ( λ ) : λ ∈ Λ } is called decomposition of S . For decomposable S we define the subspace S D by S D = w ∗ -cl lin { D T : D ∈ D ( H ) , T ∈ S} ;it is an analog of the algebra R ⊕ Λ A ( λ ) µ (d λ ) appearing in the reduction theory ofvon Neumann algebras. 3. k -hyperreflexivity We begin our investigations of k -hyperreflexivity with variants of results due toHadwin (cf. [9, Theorems 3.8 and 6.16]), Hadwin and Nordgren (cf. [7, Theorem3.6]). Originally, they concerned hyperreflexivity and property A ( r ) but the claims P. BUDZY´NSKI, K. PIWOWARCZYK AND M. PTAK are valid for k -hyperreflexivity and A /k ( r ), respectively, as well (here, and later,“hyperreflexivity” means “1-hyperreflexivity”). Since the results can be proved ina very similar fashion to the original ones, we omit the proofs. Lemma 2.
Let k ∈ N , r ∈ [1 , ∞ ) and K ∈ (0 , ∞ ) . Suppose S is a decomposableand {S ( λ ) : λ ∈ Λ } is its decomposition. Then the following holds. (i) If, for µ -a.e. λ ∈ Λ , S ( λ ) is k -hyperreflexive and κ k ( S ( λ )) ≤ K , then S D is k -hyperreflexive and κ k ( S D ) ≤ K . (ii) If, for µ -a.e. λ ∈ Λ , S ( λ ) has property A /k ( r ) , then S D has property A /k ( r ) . If V is an abelian von Neumann algebra, then by the reduction theory (cf. [17,Theorem I.2.6]), V is (up to unitary equivalence) the diagonal algebra D ( K ) corre-sponding to a direct integral decomposition of a underlying Hilbert space K . Thealgebra C I is hyperreflexive, κ ( V ) A (1) (cf. [3, Proposition 60.1]). Hence, by Lemma 2, we get the following(the part concerning hyperreflexivity is contained in [15, Theorem 3.5]). Corollary 3.
Let V be an abelian von Neumann algebra. Then V is hyperreflexive, κ ( V ) and it has property A (1) . The next lemma is an analog of [6, Proposition II.3.4]. It reveals a relationbetween direct integrals of a constant (up to the unitary equivalence) field of sub-spaces and tensor products. The proof is essentially the same as of the originalresult so we left it to the reader. A piece of notation is required: if
V ⊂ B ( H )and N ⊂ B ( K ) are weak ∗ closed subspaces, then V ⊗ N denotes the weak ∗ closedsubspace of B ( H ⊗ K ) generated by the set { A ⊗ B : A ∈ V , B ∈ N } . Lemma 4.
Let S be a weak ∗ closed subspace of B ( H ) . Let S ⊂ B ( H ) be a decom-posable subspace and {S ( λ ) : λ ∈ Λ } be its decomposition. Suppose, for every λ ∈ Λ ,there is a unitary operator U ( λ ) ∈ B ( H , H ( λ )) such that U ( λ ) S U ( λ ) − = S ( λ ) .Then there is a unitary operator U : H ⊗ H → H such that U (cid:0) D ( H ) ⊗ S (cid:1) U − = S D . We are now ready to prove k -hyperreflexivity of the tensor product of some sub-spaces. This is related to a result due to S. Rosenoer concerning hyperreflexivityof the tensor product of a hyperreflexive von Neumann algebra and the algebra ofall analytic Toeplitz operators on the Hardy space H ( D ) (cf. [16, Theorem 2]). Proposition 5.
Let
V ⊆ B ( K ) is an abelian von Neumann algebra. Let S ⊂ B ( H ) be a decomposable subspace and {S ( λ ) : λ ∈ Λ } be its decomposition. If S is k -hyperreflexive and has property A /k (1) , then every weak ∗ closed subspace S of S ⊗ V is k -hyperreflexive and κ k ( S ) κ k ( S ) . If S = S ⊗ V , then κ k ( S ) κ k ( S ) . NOTE ON k -HYPERREFLEXIVITY OF TOEPLITZ-HARMONIC SUBSPACES 5 Proof.
By the reduction theory (cf. [17, Theorem I.2.6]), V is (up to a unitaryequivalence) the diagonal algebra D ( K ) corresponding to a direct integral decom-position of the underlying Hilbert space K . In view of Lemmas 2 and 4 the subspace S ⊗ D ( K ) is k -hyperreflexive and κ k ( S ⊗ D ( K )) κ k ( S ). Lemmas 2 and 4 im-ply that S ⊗ D ( K ) has property A /k (1). If S ⊆ S ⊗ V , then S is k -hyperreflexiveand κ k ( S ) κ k ( S ) by Lemma 1. (cid:3) Corollary 6.
Let T ( D ) be weak ∗ closed subspace of all Toeplitz operators on theHardy space on the unit disc and V be an abelian von Neumann algebra. Then everyweak ∗ closed subspace T of T ( D ) ⊗ V is -hyperreflexive, κ ( T ) and it hasproperty A / (1) . If T = T ( D ) ⊗ V , then κ ( T ) Proof.
By [12, Theorem 4.1, Proposition 4.2], T ( D ) is 2-hyperreflexive, κ ( T ( D )) T ( D ) has property A / (1). Therefore, by Proposition 5, T ( D ) ⊗ V is 2-hyperreflexive, κ ( T ( D ) ⊗ V ) A / (1). If T ⊆ T ( D ) ⊗ V ,then, in view of Lemma 1, T is 2-hyperreflexive and κ ( T ) (cid:3) Corollary 7.
Let N be a von Neumann algebra with an abelian commutant and let V be an abelian von Neumann algebra. Assume that N has property A /k (1) . Thenevery is weak ∗ closed subspace S of N ⊗ V is k -hyperreflexive, κ k ( S ) and ithas property A /k (1) . If S = N ⊗ V , then κ k ( S ) .Proof. By [15, Lemma 3.1] the algebra N is k -hyperreflexive and κ k ( N )
2. Sinceit has property A /k (1), N ⊗V is k-hyperreflexive and κ k ( N ⊗V ) S ⊆ N ⊗ V , then S is k -hyperreflexive and κ k ( S )
17 by Lemma 1. (cid:3)
Now we turn our attention to the question of k -hyperreflexivity of Toeplitz-harmonic subspaces. It is well-known that the space T ( D ) of all Toeplitz operatorsacting in the Hardy space H ( D ) is a weak ∗ closed subspace generated by T nz , T ∗ z m , n, m ∈ N , i.e., T ( D ) = w ∗ -cl { p ( T z ) + q ( T z ) ∗ : p and q are analytic polynomials } , where T z is the operator of multiplication by the independent variable acting on H ( D ). For a given A ∈ B ( H ), a Toeplitz-harmonic subspace generated by A isdefined by T ( A ) = w ∗ -cl { p ( A ) + q ( A ) ∗ : p and q are analytic polynomials } . Clearly, W ( A ) ⊆ T ( A ). We have T ( H ( D )) = T ( T z ), and W ( T z ) ⊆ T ( T z ). Recallthat W ( T z ) is hyperreflexive [5, Theorem 2]. On the other hand, T ( T z ) is transitive[1, Theorem 3.1], 2-hyperreflexive (cf. [12, Corollary 4.2]) and has property A / (1)(cf. [12, Theorem 4.1]). Furthermore, for an isometry V , the algebra W ( V ) is also P. BUDZY´NSKI, K. PIWOWARCZYK AND M. PTAK hyperreflexive [11, Corollary 6]. The question arises naturally whether T ( V ) is2-hyperreflexive. As shown below, the answer is in the affirmative. Theorem 8.
Let V ∈ B ( H ) be an isometry. Then every weak ∗ closed subspace S of T ( V ) is -hyperreflexive, κ ( S ) and it has property A / (1) . It turns out that the same conclusion holds for a Toeplitz-harmonic subspacegenerated by quasinormal operator. Both the results are proved in the same way.
Theorem 9.
Suppose that T ∈ B ( H ) is a quasinormal operator. Then every weak ∗ closed subspace S of T ( T ) is -hyperreflexive, κ ( S ) and it has property A / (1) .Proof. By Brown’s result [2, Theorem 1] every quasinormal operator is unitarilyequivalent to N ⊕ ( A ⊗ S ), where N is normal, S is the unilateral shift operatorand A is positive (if T = V is an isometry, then A = I ). Since k -hyperreflexivityis kept with the same constant by unitary equivalence it is sufficient to considerthe above model. Furthermore, S ⊆ T ( T ) ⊆ N ( N ) ⊕ W ( A ) ⊗ T ( S ), where N ( N )denotes the smallest (abelian) von Neumann algebra containing N and the identityoperator I . Hence, it suffices to show that subspaces N ( N ) and W ( A ) ⊗ T ( S ) are2-hyperreflexive and have property A / (1).Since N ( N ) is a abelian von Neumann algebra, by Corollary 3 (or [3, Theorem60.14]), it is hyperreflexive, κ ( N ( N )) A / (1). As aconsequence it is 2-hyperreflexive and κ ( N ( N )) W ( A ) is an abelian von Neumann algebra. Since S is unitarilyequivalent to T z , we see that subspaces T ( S ) and T ( T z ) = T ( D ) are unitarilyequivalent. By Corollary 6, the tensor product W ( A ) ⊗ T ( S ) is 2-hyperreflexivewith constant κ ( W ( A ) ⊗ T ( S )) A / (1).In view of [12, Corollary 5.3] all of the above yields 2-hyperreflexivity of N ( N ) ⊕W ( A ) ⊗ T ( S ) with the constant less or equal to 17. Hence S is 2-hyperreflexive asbeing a subspace of the latter (see Lemma 1). Moreover, we have κ ( S ) (cid:3) References [1] E. Azoff, M. Ptak,
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Current address : Katedra Zastosowa´n Matematyki, Uniwersytet Rolniczy w Krakowie, ul.Balicka 253c, 30-198 Krak´ow, Poland
E-mail address : [email protected] Kamila Piwowarczyk, Katedra Zastosowa´n Matematyki, Uniwersytet Rolniczy wKrakowie, ul. Balicka 253c, 30-198 Krak´ow, Poland
E-mail address : [email protected] Marek Ptak, Katedra Zastosowa´n Matematyki, Uniwersytet Rolniczy w Krakowie,ul. Balicka 253c, 30-198 Krak´ow, Poland
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