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MAPS PRESERVING THE LOCAL SPECTRUM OF PRODUCT OF OPERATORS

Published online by Cambridge University Press:  17 December 2014

ABDELLATIF BOURHIM
Affiliation:
Department of Mathematics, Syracuse University, 215 Carnegie Building, Syracuse, NY 13244, USA e-mail: [email protected]
JAVAD MASHREGHI
Affiliation:
Département de mathématiques et de statistique, Université Laval, Québec, QC, G1V 0A6, Canada e-mail: [email protected]
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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 xX, let σT(x) denote the local spectrum of T at x. For two nonzero vectors x0X and y0Y, 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).

Type
Research Article
Copyright
Copyright © Glasgow Mathematical Journal Trust 2014 

References

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