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Universal interpolating sequences on spaces of analytic functions
Acta Mathematica Scientia 2011,31B(1):68–72 http://actams.wipm.ac.cn
UNIVERSAL INTERPOLATING SEQUENCES ON SPACES OF ANALYTIC FUNCTIONS∗ Dedicated to Mola Ali B.Yousefi Department of Mathematics, Payame-Noor University, Shahrake Golestan, P.O.Box 71955-1368, Shiraz, Iran E-mail: [email protected]
Abstract This article derives the relation between universal interpolating sequences and some spectral properties of the multiplication operator by the independent variable z in case the underlying space is a Hilbert space of functions analytic on the open unit disk. Key words Hilbert spaces of analytic functions; universal interpolating sequence; multiplication operator; invariant subspaces; spectral properties 2000 MR Subject Classification
1
47B38; 47A10
Introduction
In this section, we include some preparatory material that will be needed later. Let D be the open unit disk in the complex plane. The Hilbert space H under consideration satisfies the following axioms: Axiom 1 H is a subspace of the space of all analytic functions on D. Axiom 2 If f is a function in H, then so does zf . Axiom 3 For each λ ∈ D, the linear functional of evaluation at λ, eλ , is bounded on H. Axiom 4 If f is a function in H and f (λ) = 0, then there is a function g ∈ H, such that (z − λ)g = f . A Hilbert space H satisfying the above axioms is called a Hilbert space of analytic functions on D. The Hardy and Bergman spaces are examples for Hilbert spaces of analytic functions on the open unit disk. Let M be a closed subspace of H. Then, M is called invariant under an operator A ∈ B(H) if Af ∈ M for each f ∈ M. The lattice of all invariant closed subspaces of A is denoted by Lat(A). By M⊥ we denote the annihilator of M in H. If M ∈ Lat(A), then, M⊥ ∈ Lat(A∗ ). Suppose that {wn }n∈N is a sequence of distinct points in D and consider the linear transformation T : H → ∞ defined by f (wn ) Tf = . ||ewn || n∈N ∗ Received
August 28, 2007; revised June 4, 2009.
No.1
Yousefi: UNIVERSAL INTERPOLATING SEQUENCES
69
Following the interpolation theory, the sequence {wn }n∈N is called a universal interpolating sequence for H if T maps H onto 2 [1, 2]. In this article, we characterize the relation between universal interpolating sequences and some spectral properties of the shift operator, and also give some spectral properties for shift operators on invariant subspaces. This is a continuation of our work about multiplication operators on spaces of analytic functions in [3]. For some other sources on these topics one can refer to [4–11].
2
Main Results
In the following, by σ(S), σc (S), σp (S), and σap (S) we mean respectively the spectrum, compression spectrum, point spectrum, and approximate point spectrum of a bounded operator S acting on H. From now on, let H be a Hilbert space of analytic functions on the open unit disk D (satisfying the Axioms 1, 2, 3 and 4). A few comments are in order. It follows immediately from Axioms 1, 2, 3 and the Closed Graph Theorem that the multiplication by the independent variable z defines a bounded linear operator Mz on H. Also, note that, by Axiom 3, point evaluations are continuous, so the Riesz representation theorem implies that for each λ in D there is a unique function kλ ∈ H, such that eλ (f ) = f (λ) = f, kλ for all f ∈ H. The function kλ is called the reproducing kernel for the point λ. By Axiom 4, if λ ∈ D, then ran(Mz − λ) = ker eλ . Also, by Axiom 3, if λ ∈ D, then ran(Mz − λ) ⊆ ker eλ , so in Axiom 4, we only assume that ker eλ ⊆ ran(Mz − λ). In this section, we study the spectral properties of the conjugate shift operator acting in Hilbert spaces of analytic functions in the open unit disc. Such operators are very important in both complex analysis and operator theory: they deliver a universal model of contraction operator, and lead one to new intriguing problems on analytic functions. A class of such problem is related to subspaces with large codimension which are invariant with respect to operator of multiplication by z. One of basic works in this area is “ Invariant subspaces in Banach spaces of analytic functions” due to S. Richter [2], which is a pretty large work that contains a number of interesting results and indeed it is mainly of auxiliary nature. Here, we present two theorems (related to results from Section 4 in [2]) on the spectra of the adjoint of the shift, Mz∗ , restricted to the complement M⊥ of a zero based shift invariant subspace M, that is, M = {f ∈ H : f (λ) = 0 ∀λ ∈ Z}, where Z ⊆ D is a given set and M ∈ Lat(Mz ). The special case when the zero set is a universal interpolating sequence is also discussed. Indeed, in the first theorem, under the conditions M ∈ Lat(Mz ) and codM = 1, we will prove the equality of the point spectrum and the spectrum of the operator Mz∗ |M⊥ that gives additional projection on a subspace with codimension one, which was stated as Theorem 4.5 (a) in [2] about the equality of the approximate point spectrum
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and the spectrum of the operator Mz∗ |M⊥ . In our proof, we use the concept of compression spectrum and the decomposition H = [kw ] ⊕ [kw ]⊥ , where [kw ] = span{kw }. Our second theorem relates universal interpolating sequences and the inverse shift operator. Also, as a corollary, we obtain the results of Proposition 4.9 in [2] that was stated only for Bergman spaces. Note that our proofs hold generally for the Banach space of analytic functions; however, in the Hilbert space setting, the proofs turn out to be simpler. Recall that a closed subspace M ∈ Lat(Mz ) has the codimension n property if dim(M/zM) = n and in this case we write codM = n. Theorem 1 If M ∈ Lat(Mz ) and codM = 1, then, σ(Mz∗ |M⊥ ) ∩ D = σp (Mz∗ |M⊥ ) ∩ D. Proof Set Z(M) = {λ ∈ D : f (λ) = 0 ∀f ∈ M}. First, note that, if w ∈ Z(M), then, f (w) = f, kw = 0 for all f ∈ M and so kw ∈ M⊥ . But (Mz − w)∗ kw = 0, hence, w ∈ σp (Mz∗ |M⊥ ). Define Z(M)∗ = {λ : λ ∈ Z(M)}. Hence, Z(M)∗ ⊆ σap (Mz∗ |M⊥ ) ∩ D. Now, if w ∈ σap (Mz∗ |M⊥ ) ∩ D, then, there exists a sequence {hn }n∈N of unit vectors in M⊥ , such that ||(Mz − w)∗ hn || → 0. But we can write hn = cn kw + gn , where gn , kw = 0 (note that, as w ∈ D, kw is defined). As ran(Mz − w) = ker ew , ran(Mz − w) is closed and so ran(Mz − w)∗ is also closed. But ker(Mz − w)∗ = (ran(Mz − w))⊥ = [kw ], thus, (Mz − w)∗ is injective on [kw ]⊥ and hence, (Mz − w)∗ is bounded below on [kw ]⊥ . Now, as ||(Mz − w)∗ gn || → 0, we get ||gn || → 0. The sequence {cn }n∈N is clearly bounded. By passing to a subsequence, if necessary, assume that cn → c. Thus, hn → ckw and so kw ∈ M⊥ . This implies that w ∈ σp (Mz∗ |M⊥ ) ∩ D. Therefore, we obtain Z(M)∗ ⊆ σap (Mz∗ |M⊥ ) ∩ D = σp (Mz∗ |M⊥ ) ∩ D. Now, to complete the proof, we need to show that σc (Mz∗ |M⊥ ) ∩ D ⊆ Z(M)∗ . For this, let w ∈ D\ Z(M) and h ∈ ((Mz − w)∗ M⊥ )⊥ . Then, for every g ∈ M⊥ , we have (Mz − w)h, g = h, (Mz − w)∗ g = 0.
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This implies that (z−w)h ∈ M and so h ∈ M, since codM = 1. Hence, ((Mz − w)∗ M⊥ )⊥ ⊆ M and so M⊥ ⊆ (Mz − w)∗ M⊥ , which implies that w ∈ σc (Mz∗ |M⊥ ) ∩ D. This completes the proof. Corollary 2 If M ∈ Lat(Mz ) and M is cyclic, then, σ(Mz∗ |M⊥ ) ∩ D = σp (Mz∗ |M⊥ ) ∩ D. Proof As M is cyclic, codM = 1 ([2, Corollary 3.3]) and so the proof is completed. For a set Ω ⊆ C, we will write Ω− to denote the closure of Ω. Theorem 3 Let {wn }n∈N be a sequence of distinct points in D and M = {f ∈ H: f (wn ) = 0
∀n ∈ N }.
Then, the sequence {wn }n∈N forms a universal interpolating sequence for H if and only if the operator Mz∗ |M⊥ is similar to the diagonal operator ⎤ ⎡ w1 0 0 ⎥ ⎢ ⎥ ⎢ W = ⎢ 0 w2 0 ⎥ . ⎦ ⎣ .. . Proof If T : H → ∞ is defined as before in introduction, then, ker T = M = {f ∈ H: f (wj ) = 0
∀j ∈ N } =
∞
ker ewj .
j=1
So, {wj }j∈N being a universal interpolating sequence implies that T : M⊥ → 2 is one-to-one and onto, and M⊥ = span{kwj : j = 1, · · · , ∞} because f ∈ M if and only if f (wj ) = 0 for all j, and this holds if and only if f ⊥ kwj for all j (here span{.} means the closed linear span of the set {.}). Moreover, ran(T ∗ ) is closed because ran(T ) is closed. Also, T ∗ is a Banach space isomorphism between 2 and M⊥ . If {en } is the standard orthonormal basis for 2 , then, {T ∗en }n∈N is a Schauder basis for M⊥ . If f ∈ H and n ∈ N, then, we obtain kwn f (wn ) f (wk ) ∗ , en = = f, . f, T en = T f, en = ||kwk || ||kwn || ||kwn || This implies that T ∗ en = kwn /kwn . But Mz∗
kwn kwn = wn , kwn kwn
so Mz∗ is diagonal with respect to the Schauder basis {kwn /kwn }n∈N . Hence, Mz∗ |M⊥ is similar to the diagonal operator W with diagonal entries w1 , w2 , · · · and indeed Mz∗ |M⊥ T ∗ = T ∗W . Conversely, let Mz∗ |M⊥ be similar to the diagonal operator W . So, there exists an isomorphism A: 2 → M⊥ , such that Mz∗ |M⊥ A = AW.
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Set fn = Aen . Note that AW en = w n fn and Mz∗ Aen = Mz∗ fn . Hence, we get Mz∗ fn = w n fn , which implies that it should be fn = kwn /kwn . Thus, Aen = kwn /kwn and so, f (wn ) f (wk ) kwn = = , en . A∗ f, en = f, Aen = f, ||kwn || ||kwn || ||kwk || Therefore, we obtain A∗ f =
f (wn ) ||ewn ||
. n∈N
Define T = A∗ , then T : M⊥ → 2 is an isomorphism, which implies that the operator T : H → 2 given by f (wn ) Tf = ||kwn || n∈N is surjective. Thus, {wn }n∈N is a universal interpolating sequence for H as required. Corollary 4 Suppose that {wn }n∈N is a universal interpolating sequence for H. If
M= ker ewn , then, n∈N
(σ(Mz∗ |M⊥ ) ∩ D¯) = (σp (Mz∗ |M⊥ ) ∩ D¯) = (σap (Mz∗ |M⊥ ) ∩ D¯) = {wn ¯}n∈N . Remark Corollary 4 generalizes the results of Proposition 4.9 in [2] that was stated only for the Bergman spaces. References [1] Chan K C, Shields A L. Zero sets, interpolating sequences, and cyclic vectors for Dirichlet spaces. Michigan Math J, 1992, 39: 289–307 [2] Richter S. Invariant subspaces in Banach spaces of analytic functions. Trans Amer Math Soc, 1987, 304: 585–616 [3] Seddighi K, Yousefi B. On the reflexivity of operators on function spaces. Proc Amer Math Soc, 1992, 116: 45–52 [4] Shapiro H S, Shields A L. On some interpolation problems for analytic functions. Amer J Math, 1961, 83: 513–532 [5] Yousefi B. Interpolating sequence on certain Banach spaces of analytic functions. Bull Austral Math Soc, 2002, 65: 77–182 [6] Yousefi B. Multiplication operators on Hilbert spaces of analytic functions. Archiv der Mathematik, 2004, 83(6): 536–539 [7] Yousefi B, Tabatabaie B. Universal interpolating sequence on some function spaces. Czechoslovak Mathematical Journal, 2005, 55(130): 773–780 [8] Yousefi B, Foroutan S. On the multiplication operators on spaces of analytic functions. Studia Mathematica, 2005, 168(2): 187–191 [9] Yousefi B, Bagheri L. Intertwining multiplication operators on function spaces. Bulletin of the Polish Academy of Sciences Mathematics, 2006, 54(3/4): 273–276 [10] Yousefi B, Rezaei H. Hypercyclic property of weighted composition operators. Proc Amer Math Soc, 2007, 135(10): 3263–3271 [11] Yousefi B, Hesameddini E. Extension of the Beurling’s theorem. Proc Japan Acad, 2008, 84: 167–169
UNIVERSAL INTERPOLATING SEQUENCES ON SPACES OF ANALYTIC FUNCTIONS∗ Dedicated to Mola Ali B.Yousefi Department of Mathematics, Payame-Noor University, Shahrake Golestan, P.O.Box 71955-1368, Shiraz, Iran E-mail: [email protected]
Abstract This article derives the relation between universal interpolating sequences and some spectral properties of the multiplication operator by the independent variable z in case the underlying space is a Hilbert space of functions analytic on the open unit disk. Key words Hilbert spaces of analytic functions; universal interpolating sequence; multiplication operator; invariant subspaces; spectral properties 2000 MR Subject Classification
1
47B38; 47A10
Introduction
In this section, we include some preparatory material that will be needed later. Let D be the open unit disk in the complex plane. The Hilbert space H under consideration satisfies the following axioms: Axiom 1 H is a subspace of the space of all analytic functions on D. Axiom 2 If f is a function in H, then so does zf . Axiom 3 For each λ ∈ D, the linear functional of evaluation at λ, eλ , is bounded on H. Axiom 4 If f is a function in H and f (λ) = 0, then there is a function g ∈ H, such that (z − λ)g = f . A Hilbert space H satisfying the above axioms is called a Hilbert space of analytic functions on D. The Hardy and Bergman spaces are examples for Hilbert spaces of analytic functions on the open unit disk. Let M be a closed subspace of H. Then, M is called invariant under an operator A ∈ B(H) if Af ∈ M for each f ∈ M. The lattice of all invariant closed subspaces of A is denoted by Lat(A). By M⊥ we denote the annihilator of M in H. If M ∈ Lat(A), then, M⊥ ∈ Lat(A∗ ). Suppose that {wn }n∈N is a sequence of distinct points in D and consider the linear transformation T : H → ∞ defined by f (wn ) Tf = . ||ewn || n∈N ∗ Received
August 28, 2007; revised June 4, 2009.
No.1
Yousefi: UNIVERSAL INTERPOLATING SEQUENCES
69
Following the interpolation theory, the sequence {wn }n∈N is called a universal interpolating sequence for H if T maps H onto 2 [1, 2]. In this article, we characterize the relation between universal interpolating sequences and some spectral properties of the shift operator, and also give some spectral properties for shift operators on invariant subspaces. This is a continuation of our work about multiplication operators on spaces of analytic functions in [3]. For some other sources on these topics one can refer to [4–11].
2
Main Results
In the following, by σ(S), σc (S), σp (S), and σap (S) we mean respectively the spectrum, compression spectrum, point spectrum, and approximate point spectrum of a bounded operator S acting on H. From now on, let H be a Hilbert space of analytic functions on the open unit disk D (satisfying the Axioms 1, 2, 3 and 4). A few comments are in order. It follows immediately from Axioms 1, 2, 3 and the Closed Graph Theorem that the multiplication by the independent variable z defines a bounded linear operator Mz on H. Also, note that, by Axiom 3, point evaluations are continuous, so the Riesz representation theorem implies that for each λ in D there is a unique function kλ ∈ H, such that eλ (f ) = f (λ) = f, kλ for all f ∈ H. The function kλ is called the reproducing kernel for the point λ. By Axiom 4, if λ ∈ D, then ran(Mz − λ) = ker eλ . Also, by Axiom 3, if λ ∈ D, then ran(Mz − λ) ⊆ ker eλ , so in Axiom 4, we only assume that ker eλ ⊆ ran(Mz − λ). In this section, we study the spectral properties of the conjugate shift operator acting in Hilbert spaces of analytic functions in the open unit disc. Such operators are very important in both complex analysis and operator theory: they deliver a universal model of contraction operator, and lead one to new intriguing problems on analytic functions. A class of such problem is related to subspaces with large codimension which are invariant with respect to operator of multiplication by z. One of basic works in this area is “ Invariant subspaces in Banach spaces of analytic functions” due to S. Richter [2], which is a pretty large work that contains a number of interesting results and indeed it is mainly of auxiliary nature. Here, we present two theorems (related to results from Section 4 in [2]) on the spectra of the adjoint of the shift, Mz∗ , restricted to the complement M⊥ of a zero based shift invariant subspace M, that is, M = {f ∈ H : f (λ) = 0 ∀λ ∈ Z}, where Z ⊆ D is a given set and M ∈ Lat(Mz ). The special case when the zero set is a universal interpolating sequence is also discussed. Indeed, in the first theorem, under the conditions M ∈ Lat(Mz ) and codM = 1, we will prove the equality of the point spectrum and the spectrum of the operator Mz∗ |M⊥ that gives additional projection on a subspace with codimension one, which was stated as Theorem 4.5 (a) in [2] about the equality of the approximate point spectrum
70
ACTA MATHEMATICA SCIENTIA
Vol.31 Ser.B
and the spectrum of the operator Mz∗ |M⊥ . In our proof, we use the concept of compression spectrum and the decomposition H = [kw ] ⊕ [kw ]⊥ , where [kw ] = span{kw }. Our second theorem relates universal interpolating sequences and the inverse shift operator. Also, as a corollary, we obtain the results of Proposition 4.9 in [2] that was stated only for Bergman spaces. Note that our proofs hold generally for the Banach space of analytic functions; however, in the Hilbert space setting, the proofs turn out to be simpler. Recall that a closed subspace M ∈ Lat(Mz ) has the codimension n property if dim(M/zM) = n and in this case we write codM = n. Theorem 1 If M ∈ Lat(Mz ) and codM = 1, then, σ(Mz∗ |M⊥ ) ∩ D = σp (Mz∗ |M⊥ ) ∩ D. Proof Set Z(M) = {λ ∈ D : f (λ) = 0 ∀f ∈ M}. First, note that, if w ∈ Z(M), then, f (w) = f, kw = 0 for all f ∈ M and so kw ∈ M⊥ . But (Mz − w)∗ kw = 0, hence, w ∈ σp (Mz∗ |M⊥ ). Define Z(M)∗ = {λ : λ ∈ Z(M)}. Hence, Z(M)∗ ⊆ σap (Mz∗ |M⊥ ) ∩ D. Now, if w ∈ σap (Mz∗ |M⊥ ) ∩ D, then, there exists a sequence {hn }n∈N of unit vectors in M⊥ , such that ||(Mz − w)∗ hn || → 0. But we can write hn = cn kw + gn , where gn , kw = 0 (note that, as w ∈ D, kw is defined). As ran(Mz − w) = ker ew , ran(Mz − w) is closed and so ran(Mz − w)∗ is also closed. But ker(Mz − w)∗ = (ran(Mz − w))⊥ = [kw ], thus, (Mz − w)∗ is injective on [kw ]⊥ and hence, (Mz − w)∗ is bounded below on [kw ]⊥ . Now, as ||(Mz − w)∗ gn || → 0, we get ||gn || → 0. The sequence {cn }n∈N is clearly bounded. By passing to a subsequence, if necessary, assume that cn → c. Thus, hn → ckw and so kw ∈ M⊥ . This implies that w ∈ σp (Mz∗ |M⊥ ) ∩ D. Therefore, we obtain Z(M)∗ ⊆ σap (Mz∗ |M⊥ ) ∩ D = σp (Mz∗ |M⊥ ) ∩ D. Now, to complete the proof, we need to show that σc (Mz∗ |M⊥ ) ∩ D ⊆ Z(M)∗ . For this, let w ∈ D\ Z(M) and h ∈ ((Mz − w)∗ M⊥ )⊥ . Then, for every g ∈ M⊥ , we have (Mz − w)h, g = h, (Mz − w)∗ g = 0.
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This implies that (z−w)h ∈ M and so h ∈ M, since codM = 1. Hence, ((Mz − w)∗ M⊥ )⊥ ⊆ M and so M⊥ ⊆ (Mz − w)∗ M⊥ , which implies that w ∈ σc (Mz∗ |M⊥ ) ∩ D. This completes the proof. Corollary 2 If M ∈ Lat(Mz ) and M is cyclic, then, σ(Mz∗ |M⊥ ) ∩ D = σp (Mz∗ |M⊥ ) ∩ D. Proof As M is cyclic, codM = 1 ([2, Corollary 3.3]) and so the proof is completed. For a set Ω ⊆ C, we will write Ω− to denote the closure of Ω. Theorem 3 Let {wn }n∈N be a sequence of distinct points in D and M = {f ∈ H: f (wn ) = 0
∀n ∈ N }.
Then, the sequence {wn }n∈N forms a universal interpolating sequence for H if and only if the operator Mz∗ |M⊥ is similar to the diagonal operator ⎤ ⎡ w1 0 0 ⎥ ⎢ ⎥ ⎢ W = ⎢ 0 w2 0 ⎥ . ⎦ ⎣ .. . Proof If T : H → ∞ is defined as before in introduction, then, ker T = M = {f ∈ H: f (wj ) = 0
∀j ∈ N } =
∞
ker ewj .
j=1
So, {wj }j∈N being a universal interpolating sequence implies that T : M⊥ → 2 is one-to-one and onto, and M⊥ = span{kwj : j = 1, · · · , ∞} because f ∈ M if and only if f (wj ) = 0 for all j, and this holds if and only if f ⊥ kwj for all j (here span{.} means the closed linear span of the set {.}). Moreover, ran(T ∗ ) is closed because ran(T ) is closed. Also, T ∗ is a Banach space isomorphism between 2 and M⊥ . If {en } is the standard orthonormal basis for 2 , then, {T ∗en }n∈N is a Schauder basis for M⊥ . If f ∈ H and n ∈ N, then, we obtain kwn f (wn ) f (wk ) ∗ , en = = f, . f, T en = T f, en = ||kwk || ||kwn || ||kwn || This implies that T ∗ en = kwn /kwn . But Mz∗
kwn kwn = wn , kwn kwn
so Mz∗ is diagonal with respect to the Schauder basis {kwn /kwn }n∈N . Hence, Mz∗ |M⊥ is similar to the diagonal operator W with diagonal entries w1 , w2 , · · · and indeed Mz∗ |M⊥ T ∗ = T ∗W . Conversely, let Mz∗ |M⊥ be similar to the diagonal operator W . So, there exists an isomorphism A: 2 → M⊥ , such that Mz∗ |M⊥ A = AW.
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Set fn = Aen . Note that AW en = w n fn and Mz∗ Aen = Mz∗ fn . Hence, we get Mz∗ fn = w n fn , which implies that it should be fn = kwn /kwn . Thus, Aen = kwn /kwn and so, f (wn ) f (wk ) kwn = = , en . A∗ f, en = f, Aen = f, ||kwn || ||kwn || ||kwk || Therefore, we obtain A∗ f =
f (wn ) ||ewn ||
. n∈N
Define T = A∗ , then T : M⊥ → 2 is an isomorphism, which implies that the operator T : H → 2 given by f (wn ) Tf = ||kwn || n∈N is surjective. Thus, {wn }n∈N is a universal interpolating sequence for H as required. Corollary 4 Suppose that {wn }n∈N is a universal interpolating sequence for H. If
M= ker ewn , then, n∈N
(σ(Mz∗ |M⊥ ) ∩ D¯) = (σp (Mz∗ |M⊥ ) ∩ D¯) = (σap (Mz∗ |M⊥ ) ∩ D¯) = {wn ¯}n∈N . Remark Corollary 4 generalizes the results of Proposition 4.9 in [2] that was stated only for the Bergman spaces. References [1] Chan K C, Shields A L. Zero sets, interpolating sequences, and cyclic vectors for Dirichlet spaces. Michigan Math J, 1992, 39: 289–307 [2] Richter S. Invariant subspaces in Banach spaces of analytic functions. Trans Amer Math Soc, 1987, 304: 585–616 [3] Seddighi K, Yousefi B. On the reflexivity of operators on function spaces. Proc Amer Math Soc, 1992, 116: 45–52 [4] Shapiro H S, Shields A L. On some interpolation problems for analytic functions. Amer J Math, 1961, 83: 513–532 [5] Yousefi B. Interpolating sequence on certain Banach spaces of analytic functions. Bull Austral Math Soc, 2002, 65: 77–182 [6] Yousefi B. Multiplication operators on Hilbert spaces of analytic functions. Archiv der Mathematik, 2004, 83(6): 536–539 [7] Yousefi B, Tabatabaie B. Universal interpolating sequence on some function spaces. Czechoslovak Mathematical Journal, 2005, 55(130): 773–780 [8] Yousefi B, Foroutan S. On the multiplication operators on spaces of analytic functions. Studia Mathematica, 2005, 168(2): 187–191 [9] Yousefi B, Bagheri L. Intertwining multiplication operators on function spaces. Bulletin of the Polish Academy of Sciences Mathematics, 2006, 54(3/4): 273–276 [10] Yousefi B, Rezaei H. Hypercyclic property of weighted composition operators. Proc Amer Math Soc, 2007, 135(10): 3263–3271 [11] Yousefi B, Hesameddini E. Extension of the Beurling’s theorem. Proc Japan Acad, 2008, 84: 167–169