Spectral synthesis of Jordan operators
type: Item
instance of: scholarly article
title: Spectral synthesis of Jordan operators
zbMATH Open document ID: 1025.47019
author: Steven M. Seubert
published in: Journal of Mathematical Analysis and Applications
publication date: 2003-12-03T00:00:00Z
review text: Let \(X\) be a complex Banach space and \(T\) a bounded linear operator on \(X\). A vector \(x\) in \(X\) is called a root vector for \(T\) if there exists a complex number \(\lambda\) and a positive integer \(n\) such that \((T-\lambda I)^nx= 0\). We say that \(T\) is complete if the root vectors for \(T\) have dense linear span in \(X\). Let \(Y\) be a closed linear subspace of \(X\) with \(TY\subseteq Y\). If for every closed invariant subspace \(Y\) for \(T\), \(T|Y\) (the restriction of \(T\) to \(Y\)) is complete, then \(T\) is said to admit spectral synthesis. \textit{J. Wermer} [Proc. Am. Math. Soc. 3, 270-277 (1952; Zbl 0046.33704)] studied the problem of spectral synthesis for normal operators on a Hilbert space. For a review of progress on the general problem of spectral synthesis up to 1984, see \textit{N. K. Nikol'skij} [``The present state of the analysis-synthesis problem. I'', in: ``Fifteen papers on functional analysis'', Am. Math. Soc. Transl. (2) 124, 97-129 (1984; Zbl 0568.47004)]. In the present paper, the author seeks necessary and sufficient conditions for a Jordan operator to admit spectral synthesis.
Mathematics Subject Classification ID: 47B40
zbMATH DE Number: 1533190
zbMATH Keywords: bounded linear operator
zbMATH Keywords: root vector
zbMATH Keywords: spectral synthesis
zbMATH Keywords: Jordan operator
reviewed by: H. R. Dowson
MaRDI profile type: Publication
full work available at URL: jmaa.2000.6967
OpenAlex ID: W1991501672
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DOI: 10.1006/JMAA.2000.6967
description: scientific article; zbMATH DE number 1533190