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International Journal of Applied Mathematics
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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.
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[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.
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