On Hankel operators associated with a class of non-Toeplitz operators
type: Item
instance of: scholarly article
title: On Hankel operators associated with a class of non-Toeplitz operators
zbMATH Open document ID: 0731.47025
DOI: 10.1016/0022-1236(90)90025-G
published in: Journal of Functional Analysis
publication date: 1990-01-01T00:00:00Z
review text: The author considers operators on the space of all square-summable complex sequences. \((e_ n)\) denotes the standard basis. If \(\phi\) is some essentially bounded function on the unit circle having Fourier coefficients \((c_ k)\) the associated Toeplitz operator is that having the following matrix with respect to the standard basis. Set \(c_ 0\) constantly on the main diagonal, \(c_ 1\) on the first parallel to the main diagonal situated under this one, \(c_{-1}\) on the first parallel above the main diagonal and so on. The corresponding Hankel operator is associated to the Toeplitz operator in the usual way. Denote by S the classical shift operator of multiplicity 1. To any operator T the author associates the sequence \(H_ n(T)=J_ nTS^{n+1}\) where \(J_ ne_ i=e_{n-i}\) for \(0\leq i\leq n\) and \(J_ ne_ i=0\) if \(i>0\). The weak limit of the sequence \(H_ n(T)\) if it exists is denoted H(T) and is a Hankel operator. If T is Toeplitz then H(T) is the corresponding Hankel operator. A necessary and sufficient condition for the existence of H(T) is given. The existence of H(T) for a large class of non-Toeplitz operators is deduced. Operator theoretic analogs for classical results connecting Toeplitz operators to function theory are given.
Mathematics Subject Classification ID: 47B35
zbMATH DE Number: 4208881
zbMATH Keywords: Fourier coefficients
zbMATH Keywords: Toeplitz operator
zbMATH Keywords: Hankel operator
author: Avraham Feintuch
reviewed by: V. Matache
MaRDI profile type: Publication
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full work available at URL: 0022-1236(90)90025-g
OpenAlex ID: W2079833949
description: scientific article; zbMATH DE number 4208881