scholarly journals On the Norm of Certain Weighted Composition Operators on the Hardy Space

2009 ◽  
Vol 2009 ◽  
pp. 1-13 ◽  
Author(s):  
M. Haji Shaabani ◽  
B. Khani Robati

We obtain a representation for the norm of certain compact weighted composition operator on the Hardy space , whenever and . We also estimate the norm and essential norm of a class of noncompact weighted composition operators under certain conditions on and . Moreover, we characterize the norm and essential norm of such operators in a special case.

2008 ◽  
Vol 2008 ◽  
pp. 1-12 ◽  
Author(s):  
Sei-Ichiro Ueki ◽  
Luo Luo

We estimate the essential norm of a compact weighted composition operator acting between different Hardy spaces of the unit ball in . Also we will discuss a compact multiplication operator between Hardy spaces.


2020 ◽  
Vol 40 (4) ◽  
pp. 495-507
Author(s):  
Ching-on Lo ◽  
Anthony Wai-keung Loh

Let \(u\) and \(\varphi\) be two analytic functions on the unit disk \(\mathbb{D}\) such that \(\varphi(\mathbb{D}) \subset \mathbb{D}\). A weighted composition operator \(uC_{\varphi}\) induced by \(u\) and \(\varphi\) is defined on \(H^2\), the Hardy space of \(\mathbb{D}\), by \(uC_{\varphi}f := u \cdot f \circ \varphi\) for every \(f\) in \(H^2\). We obtain sufficient conditions for Hilbert-Schmidtness of \(uC_{\varphi}\) on \(H^2\) in terms of function-theoretic properties of \(u\) and \(\varphi\). Moreover, we characterize Hilbert-Schmidt difference of two weighted composition operators on \(H^2\).


2019 ◽  
Vol 99 (03) ◽  
pp. 473-484
Author(s):  
CHING-ON LO ◽  
ANTHONY WAI-KEUNG LOH

Let $u$ and $\unicode[STIX]{x1D711}$ be two analytic functions on the unit disc $D$ such that $\unicode[STIX]{x1D711}(D)\subset D$ . A weighted composition operator $uC_{\unicode[STIX]{x1D711}}$ induced by $u$ and $\unicode[STIX]{x1D711}$ is defined by $uC_{\unicode[STIX]{x1D711}}f:=u\cdot f\circ \unicode[STIX]{x1D711}$ for every $f$ in $H^{p}$ , the Hardy space of $D$ . We investigate compactness of $uC_{\unicode[STIX]{x1D711}}$ on $H^{p}$ in terms of function-theoretic properties of $u$ and $\unicode[STIX]{x1D711}$ .


2021 ◽  
Vol 29 (2) ◽  
pp. 243-250
Author(s):  
HAMID VAEZI ◽  
MOHAMAD NAGHLISAR

In this paper we consider the weighted composition operator uC_{\varphi} from Bloch-type space B^{\alpha} into Bers-type space H_{\beta}^{\infty}, in three cases, \alpha>1, \alpha=1 and \alpha<1. We give the necessary and sufficient conditions for boundedness and compactness of the above operator.


Author(s):  
Werkaferahu Seyoum ◽  
Tesfa Mengestie

AbstractFor holomorphic pairs of symbols $$(u, \psi )$$ ( u , ψ ) , we study various structures of the weighted composition operator $$ W_{(u,\psi )} f= u \cdot f(\psi )$$ W ( u , ψ ) f = u · f ( ψ ) defined on the Fock spaces $$\mathcal {F}_p$$ F p . We have identified operators $$W_{(u,\psi )}$$ W ( u , ψ ) that have power-bounded and uniformly mean ergodic properties on the spaces. These properties are described in terms of easy to apply conditions relying on the values |u(0)| and $$|u(\frac{b}{1-a})|$$ | u ( b 1 - a ) | , where a and b are coefficients from linear expansion of the symbol $$\psi $$ ψ . The spectrum of the operators is also determined and applied further to prove results about uniform mean ergodicity.


2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Flavia Colonna ◽  
Songxiao Li

We provide several characterizations of the bounded and the compact weighted composition operators from the Bloch space and the analytic Besov spaces (with ) into the Zygmund space . As a special case, we show that the bounded (resp., compact) composition operators from , , and to coincide. In addition, the boundedness and the compactness of the composition operator can be characterized in terms of the boundedness (resp., convergence to 0, under the boundedness assumption of the operator) of the Zygmund norm of the powers of the symbol.


Author(s):  
Tesfa Mengestie ◽  
Werkaferahu Seyoum

AbstractFor pairs of holomorphic maps $$(u,\psi )$$ ( u , ψ ) on the complex plane, we study some dynamical properties of the weighted composition operator $$W_{(u,\psi )}$$ W ( u , ψ ) on the Fock spaces. We prove that no weighted composition operator on the Fock spaces is supercyclic. Conditions under which the operators satisfy the Ritt’s resolvent growth condition are also identified. In particular, we show that a non-trivial composition operator on the Fock spaces satisfies such a growth condition if and only if it is compact.


2008 ◽  
Vol 19 (08) ◽  
pp. 899-926 ◽  
Author(s):  
ZE-HUA ZHOU ◽  
REN-YU CHEN

Let ϕ(z) = (ϕ1(z),…,ϕn(z)) be a holomorphic self-map of B and ψ(z) a holomorphic function on B, where B is the unit ball of ℂn. Let 0 < p, s < +∞, -n - 1 < q < +∞, q+s > -1 and α ≥ 0, this paper characterizes boundedness and compactness of weighted composition operator Wψ,ϕ induced by ϕ and ψ between the space F(p, q, s) and α-Bloch space [Formula: see text].


2008 ◽  
Vol 2008 ◽  
pp. 1-11 ◽  
Author(s):  
Stevo Stević

This paper finds some lower and upper bounds for the essential norm of the weighted composition operator fromα-Bloch spaces to the weighted-type spaceHμ∞on the unit ball for the caseα≥1.


1991 ◽  
Vol 33 (3) ◽  
pp. 275-279 ◽  
Author(s):  
James T. Campbell ◽  
Mary Embry-Wardrop ◽  
Richard J. Fleming ◽  
S. K. Narayan

In their paper [1], Campbell and Jamison attempted to give necessary and sufficient conditions for a weighted composition operator on an L2 space to be normal, and to be quasinormal. Those conditions, specifically Theorems I and II of that paper, are not valid (see [2] for precise comments on the other results in that paper). In this paper we present a counterexample to those theorems and state and prove characterizations of quasinormality (Theorem 1 below) and normality (Theorem 2 and Corollary 3 below). We also discuss additional examples and information concerning normal weighted composition operators which contribute to the further understanding of this class.


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