Operator Norms and Essential Norms of Weighted Composition Operators from Analytic Function Spaces into Zygmund-type Spa
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Complex Analysis and Operator Theory
Operator Norms and Essential Norms of Weighted Composition Operators from Analytic Function Spaces into Zygmund-type Spaces Shams Alyusof1 · Flavia Colonna2 Received: 5 December 2019 / Accepted: 18 July 2020 © Springer Nature Switzerland AG 2020
Abstract In this work, we characterize the bounded and compact weighted composition operators from a large class of Banach spaces X of analytic functions on the open unit disk into Zygmund-type spaces. Under more restrictive conditions, we provide an approximation of the essential norm of such operators. We also show that all bounded weighted composition operators from X to the little Zygmund-type space are compact and characterize such operators. We apply our results to the cases when X is the p Hardy space H p for 1 ≤ p ≤ ∞ and the weighted Bergman space Aα for α > −1 and 1 ≤ p < ∞. Keywords Weighted composition operator · Weighted Zygmund spaces · Hardy spaces · Weighted Bergman spaces Mathematics Subject Classification Primary: 47B38, 47B33; Secondary: 30H10
In memory of Carlos Berenstein who inspired an entire generation of mathematicians. Communicated by Irene Sabadini. This article is part of the topical collection “In honor of CA Berenstein and D.C. Struppa”.
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Flavia Colonna [email protected] Shams Alyusof [email protected]
1
Department of Mathematics, Al Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh, Saudi Arabia
2
Department of Mathematical Sciences, George Mason University, Fairfax, VA 22030, USA 0123456789().: V,-vol
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S. Alyusof, F. Colonna
1 Introduction Let D denote the open unit disk in the complex plane, H (D) the set of analytic functions on D, S(D) the set of analytic functions mapping D into itself, and Aut(D) the group of conformal automorphisms of D. By a Banach space of analytic functions we mean a Banach space X with norm · whose elements are analytic functions on D and whose point-evaluation functionals are bounded. Thus, for each z ∈ D, we define K (z) = sup{| f (z)| : f ∈ X , f ≤ 1} = z ,
(1)
where z denotes the point-evaluation functional at z. Then, for any f ∈ X and z ∈ D, | f (z)| ≤ f K (z).
(2)
Let μ be a positive continuous function on D, which we will call a weight. The weighted-type Banach space Hμ∞ is the space of functions f ∈ H (D) such that f Hμ∞ := sup μ(z)| f (z)| < ∞, z∈D
∞ denotes the subspace whose elements satisfy the condition with norm · Hμ∞ , and Hμ,0
lim μ(z)| f (z)| = 0.
|z|→1
The Bloch-type space Bμ is the Banach space of analytic functions f on D such that f ∈ Hμ∞ , with norm f Bμ = | f (0)| + f Hμ∞ . The little Bloch-type space ∞ . Bμ,0 is the subspace consisting of the functions f such that f ∈ Hμ,0 The weighted Zygmund-type space with weight μ is the space Zμ of functions f ∈ H (D) satisfying the condition f ∈ Hμ∞ (equivalently, f ∈ Bμ ). The norm in Zμ is defined by f Zμ = | f (0)| + | f (0)| + sup μ(z)| f (z)| = | f (0)| + f Bμ . z∈D
The little weighted Zygmund-type space with weight
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