Essential norms of weighted composition operators and Aleksandrov measures
type: Item
instance of: scholarly article
title: Essential norms of weighted composition operators and Aleksandrov measures
zbMATH Open document ID: 1231.47025
author: Pekka J. Nieminen
published in: Journal of Mathematical Analysis and Applications
publication date: 2011-07-18T00:00:00Z
review text: The purpose of this interesting paper is to improve earlier results by \textit{D. Sarason} [Lect. Notes Pure Appl. Math. 122, 545--565 (1990; Zbl 0712.47026)], \textit{J. A. Cima} and \textit{A. L. Matheson} [Pac. J. Math. 179, No.~1, 59--64 (1997; Zbl 0871.47027)] and by \textit{T. Kriete} and \textit{J. Moorhouse} [Trans. Am. Math. Soc. 359, No.~6, 2915--2944 (2007; Zbl 1115.47023)] and to present quantitative refinements that establish exact formulas for the essential and weak essential norms of weighted composition operators \(M_wC_\varphi\), acting either on the spaces \(M(\mathbb{T})\) and \(L^p(\mathbb{T})\), \(1 \leq p \leq \infty\), of complex Borel measures and \(p\)-integrable functions on the unit circle \(\mathbb{T}\), or on the Hardy space \(H^2\). Here, \(\varphi\) is an analytic self-map on the unit disc \(\mathbb{D}\) and \(w\) is a bounded Borel measurable function \(w:\mathbb{T} \rightarrow \mathbb{C}\) in the case of \(M(\mathbb{T})\) and \(w \in H^\infty\) in that of \(H^2\), under the assumption, in both cases, that \(|w|\) is continuous in the union of the closed supports of the Aleksandrov measures of \(\varphi\). The proofs are based on a general method of producing compact and weakly compact approximants for continuous linear operators \(T:L^1(X,\mu) \rightarrow L^1(Y,\nu)\) between spaces of integrable functions on metric measure spaces. The idea is originally due to Weiss and is reformulated in Section 2. Interpolation arguments and suitable test functions are also utilized in the proofs. A quantitative version of a result by \textit{E. Saksman} and the author in [J. Math. Anal. Appl. 298, No.~2, 501--522 (2004; Zbl 1072.47021)] on the (weak) essential norm of differences of composition operators is presented in Section 5. A nice example uses the obtained formula to derive explicit lower bounds of the essential norm of the difference.
reviewed by: José Bonet
Mathematics Subject Classification ID: 47B33
Mathematics Subject Classification ID: 47B07
Mathematics Subject Classification ID: 30H10
zbMATH DE Number: 5925578
zbMATH Keywords: composition operators
zbMATH Keywords: compact operators
zbMATH Keywords: essential norm
zbMATH Keywords: Aleksandrov measure
MaRDI profile type: Publication
full work available at URL: j.jmaa.2011.04.068
OpenAlex ID: W1972521399
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DOI: 10.1016/J.JMAA.2011.04.068
description: scientific article; zbMATH DE number 5925578