Analytic Toeplitz operators on the Hardy space \(H^p\): A survey
type: Item
instance of: scholarly article
title: Analytic Toeplitz operators on the Hardy space \(H^p\): A survey
zbMATH Open document ID: 1039.47016
author: Dragan Vukotić
published in: Bulletin of the Belgian Mathematical Society - Simon Stevin
publication date: 2004-02-03T00:00:00Z
review text: This is a neat overview of results on analytic Toeplitz operators -- that~is, operators \(M_\phi\) of multiplication by analytic functions~\(\phi\) -- on~the Hardy spaces~\(H^p\), \(1\leq p<\infty\), of~the unit circle. The~topics addressed include criteria for boundedness and identification of the operator norm \(\| M_\phi\| \) with \(\| \phi\| _\infty\), the characterization of \(\{M_\phi\), \(\phi\in H^\infty\}\) as the commutant of~\(M_z\), compactness and the equality of the essential norm \(\| M_\phi\| _e\) and~\(\| \phi\| _\infty\), a description of the spectrum and the essential spectrum, criteria for closedness of the range, and Fredholmness and index theory. While all these results are very classical for the Hilbert space case \(p=2\), the information on the general case \(1\leq p<\infty\) has hitherto been rather scattered, and it is therefore very useful to have all these facts gathered in one place.
reviewed by: Miroslav Engliš
Mathematics Subject Classification ID: 47B35
Mathematics Subject Classification ID: 30H05
Mathematics Subject Classification ID: 47-02
zbMATH DE Number: 2031057
zbMATH Keywords: Hardy space
zbMATH Keywords: analytic Toeplitz operator
zbMATH Keywords: essential norm
zbMATH Keywords: spectrum
zbMATH Keywords: Fredholm operator
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 2031057