Spectrum-preserving elementary operators on \(B(X)\)
type: Item
instance of: scholarly article
title: Spectrum-preserving elementary operators on \(B(X)\)
zbMATH Open document ID: 0918.47006
published in: Chinese Annals of Mathematics. Series B
publication date: 1999-08-19T00:00:00Z
review text: Let \(X\) be an infinite-dimensional complex Banach space and \(B(X)\) the Banach algebra of all bounded linear operators on \(X\). For \(T\in B(X)\), \(\sigma(T)\), as usual, will denote the spectrum of \(T\). Let \(\Phi\) be a linear map from \(B(X)\) into itself. \(\Phi\) is spectrum-preserving if \(\sigma(\Phi(T))= \sigma(T)\) for all \(T\in B(X)\); \(\Phi\) is spectrum-compressing if \(\sigma(\Phi(T))\subseteq \sigma(T)\) for all \(T\in B(X)\). It is clear that if \(\Phi\) is unital (i.e., \(\Phi(I)= I\)), then \(\Phi\) is spectrum-preserving (spectrum compressing) if and only if \(\Phi\) preserves invertibility in both directions (preserves invertibility), i.e., \(\Phi(T)\) is invertible if and only if \(T\) is (\(\Phi(T)\) is invertible if \(T\) is). Spectrum-preserving linear maps have been studied by many authors. In fact, this is one of the so-called linear preserver problems. It is interesting to ask if one can give a characterization for the structure of spectrum-preserving linear maps which are not surjective. But this question seems very difficult to answer. An important class of linear maps on \(B(X)\) which contains many non-surjective maps is the class of elementary operators. Recall that \(\Phi\) is called an elementary operator if there exist operators \(A_1,\dots, A_n\), \(B_1,\dots, B_n\) in \(B(X)\) such that \[ \Phi(T)= \sum^n_{i=1} A_iTB_i\quad\text{for all }T\in B(X). \] The number \(\ell(\Phi)= \inf\{n; \Phi(.)= \sum^n_{i=1} A_i(.)B_i\}\) is called the length of \(\Phi\). The main result of this paper is the following theorem: Theorem. Let \(\Phi(.)= A_1(.)B_1+ A_2(.)B_2\) be an elementary operator of length 2 with \(I\in R(\Phi)\). Then the following statements are equivalent. (i) \(\Phi\) is spectrum-compressing; (ii) \(\Phi\) is spectrum-preserving; (iii) \(A= (A_1,A_2)\) is invertible with \(A^{-1}= \left(\begin{smallmatrix} B_1\\ B_2\end{smallmatrix}\right)\); (iv) There exists an invertible operator \(A\in B(X\oplus X,X)\) such that \…
Mathematics Subject Classification ID: 47A10
Mathematics Subject Classification ID: 47B47
zbMATH DE Number: 1229550
zbMATH Keywords: elementary operator
zbMATH Keywords: spectrum-preserving linear maps
zbMATH Keywords: spectrum-compressing
zbMATH Keywords: linear preserver problems
zbMATH Keywords: invertible operator
author: Jinchuan Hou
reviewed by: Mahmoud Kutkut
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1229550