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MAPS PRESERVING THE LOCAL SPECTRUM OF PRODUCT OF OPERATORS

Part of: General theory of linear operators Special classes of linear operators

Published online by Cambridge University Press:  17 December 2014

ABDELLATIF BOURHIM  and
JAVAD MASHREGHI
ABDELLATIF BOURHIM
Affiliation:
Department of Mathematics, Syracuse University, 215 Carnegie Building, Syracuse, NY 13244, USA e-mail: abourhim@syr.edu
JAVAD MASHREGHI
Affiliation:
Département de mathématiques et de statistique, Université Laval, Québec, QC, G1V 0A6, Canada e-mail: javad.mashreghi@mat.ulaval.ca
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Abstract

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Let X and Y be infinite-dimensional complex Banach spaces, and ${\mathcal B}$(X) (resp. ${\mathcal B}$(Y)) be the algebra of all bounded linear operators on X (resp. on Y). For an operator T ∈ ${\mathcal B}$(X) and a vector x ∈ X, let σT(x) denote the local spectrum of T at x. For two nonzero vectors x0 ∈X and y0 ∈ Y, we show that a map ϕ from ${\mathcal B}$(X) onto ${\mathcal B}$(Y) satisfies

$\sigma_{\varphi(T)\varphi(S)}(y_0)~=~\sigma_{TS}(x_0),~(T,~S\in{\mathcal B}(X)),$
if and only if there exists a bijective bounded linear mapping A from X into Y such that Ax0 = y0 and either ϕ(T) = ATA−1 or ϕ(T) = -ATA−1 for all T ∈ ${\mathcal B}$(X).

MSC classification

Primary: 47B49: Transformers, preservers (operators on spaces of operators)
Secondary: 47A10: Spectrum, resolvent 47A11: Local spectral properties

Information

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 57 , Issue 3 , September 2015 , pp. 709 - 718
Copyright
Copyright © Glasgow Mathematical Journal Trust 2014 

References

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