Algebra descent spectrum of operators
type: Item
instance of: scholarly article
title: Algebra descent spectrum of operators
zbMATH Open document ID: 1205.47005
author: Amin Kaidi
author: A. Haily
author: Ángel Rodríguez Palacios
published in: Israel Journal of Mathematics
publication date: 2010-12-13T00:00:00Z
review text: The descent \(d(T)\) of a linear operator \(T\) acting on a vector space \(X\) over a field \(K\) is defined by \(d(T):= \min\{n \in \mathbb{N}\cup\{0\}: R(T^n)=R(T^{n+1})\}\) with the convention that \(\min \emptyset = \infty\). Here, \(R(\cdot)\) denotes the range. The descent spectrum of \(T\) is the set \(\sigma_{\text{desc}}(T):= \{\lambda\in K: d(T -\lambda) = \infty\}\). Clearly, \(\sigma_{\text{desc}}(T)\) is a subset of the usual spectrum \(\sigma(T):=\{\lambda \in K : T -\lambda \text{~is~not~bijective} \}\). Also, \(\sigma_{\text{desc}}(T,B(X)) := \sigma_{\text{desc}}(TM_T)\), with \(M_T\) the multiplication operator \(M_T(S) = TS\) on \(B(X)\). The ``descent spectrum equality'' (in short, DSE) means that for every \(T\) in the algebra \(B(X)\) of all bounded linear operators on \(X\), the equality \(\sigma_{\text{desc}}(T)=\sigma_{\text{desc}}(T, B(X))\) holds. The authors investigate the interesting question ``Does every Banach space \(X\) satisfy the DSE?''. They prove that the DSE is fulfilled by \(\ell_1(I)\) and \(\ell_2(I)\) for any set \(I\), but not by \(\ell_p(I)\) for any infinite set \(I\) and any \(p\neq 2\) with \(1 < p \leq \infty\). They show that the DSE is also fulfilled by all Banach spaces which are not isomorphic to any of their proper quotients. Actually, a Banach space is not isomorphic to any of its proper quotients if and only if it is not isomorphic to any of its proper complemented subspaces and satisfies the DSE. Another characterization of Banach spaces which are not isomorphic to any of their proper quotients is the one asserting that they are precisely those Banach spaces \(X\) such that, for every \(T \in B(X)\), the interior of \(\sigma(T)\) is contained in \(\sigma_{\text{desc}}(T )\).
reviewed by: Mohammad Sal Moslehian
Mathematics Subject Classification ID: 47A10
Mathematics Subject Classification ID: 46B03
zbMATH DE Number: 5826053
zbMATH Keywords: descent spectrum equality
zbMATH Keywords: spectrum
zbMATH Keywords: descent spectrum
zbMATH Keywords: Banach space
zbMATH Keywords: sequence space
MaRDI profile type: Publication
full work available at URL: s11856-010-0050-9
OpenAlex ID: W2108226591
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DOI: 10.1007/S11856-010-0050-9
description: scientific article; zbMATH DE number 5826053