Rocky Mountain Journal of Mathematics

Composition operators on weighted Hardy spaces

Waleed Al-Rawashdeh

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Abstract

Suppose $\varphi$ is an analytic self-map of open unit disk $\mathbf{D}$ and $\psi$ is an analytic function on $\mathbf{D}$. Then a weighted composition operator induced by $\varphi$ with weight $\psi$ is given by $(W_{\psi, \varphi}f)(z)= \psi(z)f(\varphi(z))$, for $z \in \mathbf{D}$ and $f$ analytic on $\mathbf{D}$. Necessary and sufficient conditions are given for the boundedness and compactness of the weighted composition operators $W_{\psi, \varphi}$. In terms of fixed points in the closed unit disk $\overline{\mathbf{D}}$, conditions under which $W_{\psi, \varphi}$ is compact are given. Necessary conditions for the compactness of $C_{\varphi}$ are given in terms of the angular derivative $\varphi^{\prime}(\zeta)$ where $\zeta$ is on the boundary of the unit disk. Moreover, we present sufficient conditions for the membership of composition operators in the Schatten $p$-class $S_{p}(H^s(\beta_1), H^q(\beta_2))$, where the inducing map has supremum norm strictly smaller than~$1$.

Article information

Source
Rocky Mountain J. Math., Volume 44, Number 4 (2014), 1053-1072.

Dates
First available in Project Euclid: 31 October 2014

Permanent link to this document
https://projecteuclid.org/euclid.rmjm/1414760942

Digital Object Identifier
doi:10.1216/RMJ-2014-44-4-1053

Mathematical Reviews number (MathSciNet)
MR3274337

Zentralblatt MATH identifier
1320.47034

Subjects
Primary: 47B38: Operators on function spaces (general)
Secondary: 47B15: Hermitian and normal operators (spectral measures, functional calculus, etc.) 47B33: Composition operators

Keywords
Weighted composition operators compact operator angular derivative Schatten $p$-class weighted Hardy spaces

Citation

Al-Rawashdeh, Waleed. Composition operators on weighted Hardy spaces. Rocky Mountain J. Math. 44 (2014), no. 4, 1053--1072. doi:10.1216/RMJ-2014-44-4-1053. https://projecteuclid.org/euclid.rmjm/1414760942


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