The essential spectrum of AM-compact operators
type: Item
instance of: scholarly article
title: The essential spectrum of AM-compact operators
zbMATH Open document ID: 0746.47013
DOI: 10.1016/0019-3577(91)90002-O
published in: Indagationes Mathematicae. New Series
publication date: 1992-06-25T00:00:00Z
review text: The paper is devoted to the construction of a representation space for the essential spectrum of an operator in the category of Banach lattices. If \(E\) is a Banach lattice, \(s(E)\) denotes the space of those sequences \((x_ n)\) in \(E\) such that its range \(\{x_ n:\;n\in N\}\) is almost order bounded. The quotient space \(\ell_{\infty}(E)/s(E)=\tilde E\) is a Banach lattice, too. An operator \(T: E\to E\) is called quasi regular if \(T\) maps order bounded sets into almost order bounded sets. The authors show that \({\mathcal L}^{qr}=\{\) quasi-regular operators from \(E\) into \(E\}\) is a closed subalgebra of \({\mathcal L}(E)\) and that any \(T\in{\mathcal L}^{qr}(E)\) induces an operator \(\tilde T\) in \({\mathcal L}(\tilde E)\) by the formula \(\tilde T[(x_ n)+s(E)]:=(Tx_ n)+s(E)\). Then they consider \(AM\)-compact operators (i.e. operators mapping order bounded sets onto relatively compact sets). Since each \(AM\)-compact operator \(T\) is quasi- regular, \(T\) induces a \(\tilde T\in{\mathcal L}(\tilde E)\); furthermore \(T(s(E))\subset m(E)=\{(x_ n)\subset E\) that have a relatively compact range\(\}\). This inclusion is basic to show the following result: Theorem 1.5. Let \(E\) be a Banach lattice and \(T\in{\mathcal L}(E)\) be an \(AM\)-compact operator. Then (i) \(\sigma_ p(\tilde T)-\{0\}=\sigma_ p(P(T))-\{0\}\), (ii) if \(\lambda\neq 0\), then \(\lambda I-\tilde T\) is surjective if and only if \(\lambda I-P(T)\) is surjective, (iii) \(\sigma(\tilde T)-\{0\}=\sigma(P(T))-\{0\}\), (iv) \(\sigma^ +_{ess}(T)-\{0\}=\sigma_ p(\tilde T)-\{0\}\), \(\sigma^ -_{ess}(T)-\{0\}=\{\lambda\neq 0:\;\lambda I-\tilde T\text{ is not surjective}\}\) and \(\sigma_{ess}(T)-\{0\}=\sigma(\tilde T)- \{0\}\) (where \(\sigma(T)\) denotes the spectrum of \(T\), \(\sigma_ p(T)\) its point spectrum, \(\sigma_{ess}(T)\) the essential spectrum of \(T\), \(P(T)\) the bounded linear operator induced from \(\ell_ \infty(E)/m(E)\) to itself by \(T\in{\mathcal L}(E)\) and defined b…
reviewed by: G. Emmanuele
Mathematics Subject Classification ID: 47B60
Mathematics Subject Classification ID: 47A10
Mathematics Subject Classification ID: 47B65
Mathematics Subject Classification ID: 46B42
zbMATH DE Number: 11706
zbMATH Keywords: sequentially semicompact operators
zbMATH Keywords: essential spectrum
zbMATH Keywords: category of Banach lattices
zbMATH Keywords: order bounded sets
zbMATH Keywords: quasi-regular operators
zbMATH Keywords: \(AM\)- compact operators
zbMATH Keywords: peripherical essential spectrum
zbMATH Keywords: measure of non- semicompactness
author: F. Andreu
author: V. Caselles
author: Josep Martínez
author: J. M. Mazón
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 11706