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International Journal of Applied Mathematics ————————————————————– Volume 25 No. 2 2012, 299-303 REFLEXIVITY OF MULTIPLICATION OPERATORS ON WEIGHTED HARDY SPACES Bahmann Yousefi1 § , Ali Iloon Kashkooly2 1 Department of Mathematics Payame Noor University P.O. Box 19395-3697, Tehran, IRAN e-mail: b yousefi@pnu.ac.ir 2 Department of Mathematics Yasouj University Yasouj, IRAN e-mail: [email protected] Abstract: We give sufficient conditions under which the powers of the multiplication operator are reflexive. AMS Subject Classification: 47B37, 46A25, 47L10 Key Words: Banach and Hardy spaces, Laurent series associated with a sequence β, weak operator topology, reflexive operator, multiplication operator 1. Introduction In this section we include some preparatory material which is needed later. Let X be a reflexive Banach space. For the algebra B(X) of all bounded operators on the Banach space X, the weak operator topology is the one in which a net Aα converges to A if Aα x → Ax weakly, x ∈ X. Recall that if A ∈ B(X), then Lat(A) is by definition the lattice of all invariant subspaces of A, and AlgLat(A) is the algebra of all operators B in B(X) such that Lat(A) ⊂ Lat(B). An operator A in B(X) is said to be reflexive if AlgLat(A) = W (A), where W (A) is the smallest subalgebra of B(X) that contains A and the identity I and is Received: March 28, 2012 § Correspondence author c 2012 Academic Publications 300 B. Yousefi, A.I. Kashkooly closed in the weak operator topology. For some sources of reflexivity see [1–6]. Let β = {β(n)}∞ n=−∞ be a sequence of positive numbers satisfying β(0) = 1. If 1 ≤ p < ∞, the space Lp (β) consists of all formal Laurent series f (z) = ∞ |fˆ(n)|p β(n)p is finite. fˆ(n)z n such that the norm f p = f p = ∞ β n=−∞ n=−∞ Lp (β) only contains formal power When n just runs over N ∪ {0}, the space ∞ fˆ(n)z n and it is usually denoted by H p (β). If p = 2, such series f (z) = n=0 spaces were introduced by Allen L. Shields [1] to study weighted shift operators. Let fˆk (n) = δk (n). So fk (z) = z k and then {fk }k∈Z is a basis for Lp (β) such that fk = β(k). Now consider Mz , the operator of multiplication by z on ∞ fˆ(n)z n+1 where f (z) = fˆ(n)z n ∈ Lp (β). In Lp (β): (Mz f )(z) = ∞ n=−∞ n=−∞ other words, (Mz f )ˆ(n) = fˆ(n − 1) for all n ∈ Z. Clearly Mz shifts the basis {fk }k . The operator Mz is bounded if and only if {β(k + 1)/β(k)}k is bounded and in this case Mzn = supk [β(k + n)/β(k)] for all n ∈ N ∪ {0}. Clearly Mz is invertible if and only if β(k)/β(k + 1) is bounded. We denote the set of multipliers {ϕ ∈ Lp (β) : ϕLp (β) ⊆ Lp (β)} by Lp∞ (β) and the linear operator of multiplication by ϕ on Lp (β) by Mϕ . Also the set of p (β). multipliers on H p (β) is denoted by H∞ We say that a complex number λ is a bounded point evaluation on Lp (β) if the functional e(λ) : Lp (β) −→ C defined by e(λ)(f ) = f (λ) is bounded. p By the same method used in [2] we can see that Lp (β)∗ = Lq (β q ), where p ˆ 1 1 ĝ(n)z n ∈ Lq (β q ), f (n)z n ∈ Lp (β) and g(z) = p + q = 1. Also if f (z) = n n then clearly < f, g >= n fˆ(n)ĝ(n)β(n)p . For a good source in formal power series, we refer the reader to papers [7–10]. 2. Main Results In this section we give sufficient conditions for reflexivity of the powers of the multiplication operator by the independent variable z, Mz , acting on Banach spaces of formal series. The following theorem extends the results obtained by Shields (for the case p = 2) in [1] and due to similarity, we omit the proof. For each ϕ ∈ Lp∞ (β) put: Pn (ϕ) = n (1 − k=0 k )ϕ̂(k)z k , n ≥ 0. n+1 REFLEXIVITY OF MULTIPLICATION... 301 Theorem 2.1. If ϕ ∈ Lp∞ (β) ∩ H(Ω11 ), then MPn (ϕ) −→ Mϕ in the weak operator topology. In the following theorem we use the notations: −1 r01 = limβ(−n) n 1 r11 = limβ(n) n Ω1 = Ω01 ∩ Ω11 . , Ω01 = {z ∈ C : |z| > r01 } , Ω11 = {z ∈ C : |z| < r11 } Note that if r01 < r11 , each point of Ω1 is a bounded point evaluation on Lp (β). Theorem 2.2. Let Mz be invertible on Lp (β) and r01 < r11 . If there exists c > 0 such that Ms ≤ csΩ1 for all Laurent polynomials s, then Mz is reflexive. Proof. Let A ∈ AlgLat(Mz k ). Since Lat(Mz ) ⊂ Lat(Mz k ), thus Lat(Mz ) ⊂ Lat(A). This implies that A ∈ AlgLat(Mz ). By the same method used in the proof of Theorem 1 in [2] we can see that each point of Ω1 is a bounded point evaluation on Lp (β). Since Mz∗ e(λ) = λe(λ) for all λ in Ω1 , the one dimensional span of e(λ) is invariant under Mz∗ . Therefore it is invariant under A∗ and we write A∗ e(λ) = ϕ(λ)e(λ), λ ∈ Ω1 . So < Af, e(λ) >=< f, A∗ e(λ) >= ϕ(λ)f (λ) for all f ∈ Lp (β) and λ ∈ Ω1 . This implies that A = Mϕ and ϕ ∈ Lp∞ (β). p ⊂ H ∞ (Ω1 ), by the same lemma in [1] we can write Now since ∞ϕ ∈ L∞ (β) n ϕ(z) = n=−∞ ϕ̂(n)z = ϕ1 (z) + ϕ2 (z) where ϕ1 (z) = ϕ2 (z) = ∞ ϕˆ1 (n)z n ∈ H ∞ (Ω11 ), n=0 −1 ϕˆ2 (n)z n ∈ H ∞ (Ω01 ). n=−∞ Now, first we show that ϕ2 ≡ c, a constant. To see this, note that Lp (β) ∈ Lat(Mz ), so Lp (β) ∈ Lat(A) and also Lp (β) ∈ Lat(Mϕ1 ). Hence ϕ2 = ϕ − ϕ1 = A1 − Mϕ1 1 ∈ Lp (β). If ϕ2 = c, then ϕ̂2 (k) = 0 for some k < 0. Since ϕ2 ∈ H ∞ (Ω01 ), there is a sequence of polynomials in 1z , {sn ( 1z )}n , uniformly bounded on Ω01 and converging pointwise to ϕ2 (z). Therefore sn (Mz−1 ) −→ Mϕ2 in the weak operator topology. To see this, note that 1 Ms∗n ( 1 ) e(λ) = sn ( )e(λ) −→ ϕ2 (λ)e(λ) = Mϕ∗2 e(λ). z λ 302 B. Yousefi, A.I. Kashkooly Hence Ms∗ 1 n( z ) f −→ Mϕ∗2 f for every f in the linear span of {e(λ) : λ ∈ Ω01 } p that is dense in Lq (β q ). But ϕ2 ∈ Lp∞ (β), thus by using the assumption we get 1 Msn ( 1 ) ≤ csn ( )Ω01 . z z Therefore {Msn ( 1 ) }n is uniformly bounded and hence z Ms∗n ( 1 ) f −→ Mϕ∗2 f z for every f ∈ Lp (β). We have actually shown that sn (Mz−1 ) −→ Mϕ2 in the strong operator topology. Therefore < sn (Mz−1 )1, fk >−→< Mϕ2 1, fk > where fk (z) = z k . That is sn ( 1z ˆ)(k) −→ ϕ̂2 (k) as n −→ ∞. This is a contradiction, since the left hand side is zero and the right hand side is nonzero. Hence ϕ2 is a constant, and so Mϕ = Mϕ1 + cI. Since ϕ1 ∈ Lp∞ (β) ∩ H(Ω11 ), then by Theorem 2.1, MPn (ϕ1 ) −→ Mϕ1 in the weak operator topology. Hence Mϕ1 ∈ W (Mz ). This implies that Mϕ = Mϕ1 + cI ∈ W (Mz ) and so the proof is complete. References [1] A.L. Shields, Weighted shift operators and analytic functions theory, Math. Surveys, A.M.S. Providence, 13 (1974), 49-128. [2] B. Yousefi, On the space p (β), Rend. Circ. Mat. Palermo, 49 (2000), 115-120. [3] B. Yousefi, Unicellularity of the multiplication operator on Banach spaces of formal power series, Studia Mathematica, 147, No. 3 (2001), 201-209. [4] B. Yousefi, Bounded analytic structure of the Banach space of formal power series, Rend. Circ. Mat. Palermo, Serie II, LI (2002), 403-410. [5] B. Yousefi, S. Jahedi, Composition operators on Banach spaces of formal power series, Bollettino Della Unione Matematica Italiano, (8) 6-B (2003), 481-487. REFLEXIVITY OF MULTIPLICATION... 303 [6] B. Yousefi, Strictly cyclic algebra of operators acting on Banach spaces H p (β), Czechoslovak Mathematical Journal, 54, No. 129 (2004), 261-266. [7] B. Yousefi, On the eighteenth question of Allen Shields, International Journal of Mathematics, 16, No. 1 (2005), 1-6. [8] B. Yousefi, S. Jahedi, Reflexivity of the multiplication operator on the weighted Hardy spaces, Southeast Asian Bulletin of Mathematics, 31 (2007), 163-168. [9] B. Yousefi, J. Doroodgar, Reflexivity on Banach spaces of analytic functions, Journal of Mathematical Extension, 3, No. 1 (2009), 87-93. [10] B. Yousefi, A. Khaksari, Multiplication operators on analytic functional spaces, Taiwanese Journal of Mathematics, 13, No. 4 (2009), 1159-1165. 304