Toeplitz and Hankel operators on the Paley-Wiener space
type: Item
instance of: scholarly article
title: Toeplitz and Hankel operators on the Paley-Wiener space
zbMATH Open document ID: 0634.47024
DOI: 10.1007/BF01199078
published in: Integral Equations and Operator Theory
publication date: 1987-01-01T00:00:00Z
review text: Let \(I=[-\pi,\pi]\). For a function f defined on I we denote by \(\chi^ f\) the function on R which agrees with f on I and vanishes on \(R\setminus I\). Denote the Fourier transform of g by \({\mathcal F}g=\hat g\) and the inverse Fourier transform of f by \({\mathcal F}^{-1}f=f^{\vee}\). The Paley-Wiener space, PW, is defined to be the image of \(L^ 2(I)\) under \({\mathcal F}^{-1}\). We norm PW be requiring \({\mathcal F}\) to be an isometry. The Toeplitz operator with ymbol \(\phi\) is the operator \(T_{\phi}\) with domain PW given by \(T_{\phi}(f)=P(\phi f)\) where P denotes the orthogonal projection from \(L^ 2\) to PW. The Hankel operator with symbol \(\phi\) is the operator \(H_{\phi}\) defined by \(H_{\phi}(f)=P(\phi \bar f).\) For any z in C and any t in R set \(f_ z=f_ z(t)=\chi_{2I}(t)e^{izt}\). The author shows that finite sums \(\phi =\sum c_ if^{\vee}_{z_ i}\) give finite rank operators and those are essentially the only finite rank \(H_{\phi}\). When \(\phi^{\wedge}\) is supported in \(R_+\), \(T_{\phi}\) has been studied by \textit{D. Sarason} [Trans. Am. Math. Soc. 127, 179-203 (1967; Zbl 0145.393)] and others. Pick and fix \(\psi_ L\) in \(C_ 0^{\infty}\) which has \(\sup p(\psi_ L)\subset [-4\pi,-\pi /2]\), and \(\psi_ L(x)=1\) for x in [- 3\(\pi\),-\(\pi\) ]. Set \(\psi_ R(x)=\psi_ L(-x)\) and define \(\psi_ c\) to be off 2I and \(\psi_ c=1-\psi_ R-\psi_ L\) on I. Define \(\phi_ L\) by \(\phi_ L=(\psi_ L\phi^{\wedge})^{\vee}\) and set \(T_ L=T_{\phi_ L}\). Let \(P_+\) be the orthogonal projection from \(L^ 2\) to the associated Hardy space \(H^ 2\), \(P_-=I-P_+\) and \(\theta =e^{i\pi x}\). The author shows that \(\| T_{\phi}\| \cong \| P_-({\bar \theta}^ 2\phi_ R)\|_*+\| P_+(\theta^ 2\phi_ L)\|_*+\| \phi_ c\|_{\infty}\) where \(\|\|_*\) denotes BMO norm. He gives characterizations of the symboles of \(T_{\phi}\) in the compact ideal or the Schatten-von Neumann ideals. Moreover he studies the commutator of multiplication operators with the discr…
reviewed by: Takahiko Nakazi
Mathematics Subject Classification ID: 47B35
Mathematics Subject Classification ID: 47B10
Mathematics Subject Classification ID: 47B06
Mathematics Subject Classification ID: 46J15
Mathematics Subject Classification ID: 42B30
Mathematics Subject Classification ID: 30D15
zbMATH DE Number: 4032430
zbMATH Keywords: Toeplitz operator
zbMATH Keywords: inverse Fourier transform
zbMATH Keywords: Paley-Wiener space
zbMATH Keywords: Hankel operator
zbMATH Keywords: finite rank operators
zbMATH Keywords: Hardy space
zbMATH Keywords: BMO
zbMATH Keywords: compact ideal
zbMATH Keywords: Schatten-von Neumann ideals
zbMATH Keywords: commutator of multiplication operators with the discrete Hilbert transform
author: Richard Rochberg
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01199078
OpenAlex ID: W2323096788
description: scientific article; zbMATH DE number 4032430