Compact weighted composition operators on function algebras
type: Item
instance of: scholarly article
title: Compact weighted composition operators on function algebras
zbMATH Open document ID: 0663.47021
author: H. Takagi
published in: Tokyo Journal of Mathematics
publication date: 1988-01-01T00:00:00Z
review text: Let \({\mathcal M}_ A\) denote the maximal ideal space of the uniform algebra A, let \({\mathcal M}^{\infty}_ A={\mathcal M}_ A\cup \{0\}\) and let {\^{\ }} denote the Gelfand transform. A continuous linear oprator T from A into A is called a weighted composition operator if there exists \(u\in A\) and a continuous mapping \(\phi\) from support(u) into \({\mathcal M}^{\infty}_ A\) such that \[ Tf(x)=u(x)\hat f(\phi (x))\quad if\quad x\in \sup port(u)\quad and\quad =0\quad if\quad x\not\in \sup port(u). \] In this article the author investigates conditions under which weighted composition operators are compact. Using Gleason parts and an analyticity criterion due to \textit{S. Ohno} and \textit{J. Wada} [Tokyo J. Math. 4, 105- 112 (1981; Zbl 0471.46035)] the author characterizes the compact weighted composition operators of \(H^{\infty}(D)\) and recovers known results for such operators on \({\mathcal C}(X)\) and the disc algebra.
Mathematics Subject Classification ID: 47B38
Mathematics Subject Classification ID: 46J10
Mathematics Subject Classification ID: 47B06
zbMATH DE Number: 4084409
zbMATH Keywords: maximal ideal space
zbMATH Keywords: uniform algebra
zbMATH Keywords: Gelfand transform
zbMATH Keywords: weighted composition operator
zbMATH Keywords: Gleason parts
reviewed by: Seán Dineen
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.3836/tjm/1270134266
OpenAlex ID: W1970422127
DOI: 10.3836/TJM/1270134266
description: scientific article; zbMATH DE number 4084409