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MAPS PRESERVING THE LOCAL SPECTRUM OF PRODUCT OF OPERATORS

Part of: Special classes of linear operators General theory 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
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 57 , Issue 3 , September 2015 , pp. 709 - 718
Copyright
Copyright © Glasgow Mathematical Journal Trust 2014 

References

REFERENCES

1.Aiena, P., Fredholm and local spectral theory, with applications to multipliers (Kluwer, Dordrecht, 2004).Google Scholar
2.Bhatia, R., Šemrl, P. and Sourour, A., Maps on matrices that preserve the spectral radius distance, Stud. Math. 134 (2) (1999), 99–110.CrossRefGoogle Scholar
3.Bourhim, A. and Mashreghi, J., Local spectral radius preservers, Integral Equations Operator Theory 76 (1) (2013), 95–104.CrossRefGoogle Scholar
4.Bourhim, A. and Mashreghi, J., Maps preserving the local spectrum of triple product of operators, Linear Multilinear Algebra, to appear.Google Scholar
5.Bourhim, A. and Ransford, T., Additive maps preserving local spectrum Integral Equations Operator Theory 55 (2006) 377–385.CrossRefGoogle Scholar
6.Bračič, J. and Müller, V., Local spectrum and local spectral radius of an operator at a fixed vector, Stud. Math. 194 (2) (2009), 155–162.CrossRefGoogle Scholar
7.Chan, J. T., Li, C. K. and Sze, N. S., Mappings preserving spectra of products of matrices, Proc. Am. Math. Soc. 135 (2007), 977–986.CrossRefGoogle Scholar
8.Costara, C., Linear maps preserving operators of local spectral radius zero, Integral Eqns Operator Theory 73 (1) (2012), 7–16.CrossRefGoogle Scholar
9.Costara, C., Automatic continuity for linear surjective mappings decreasing the local spectral radius at some fixed vector, Arch. Math. 95 (6) (2010), 567–573.CrossRefGoogle Scholar
10.Cui, J. L. and Hou, J. C., Maps leaving functional values of operator products invariant Linear Algebra Appl. 428 (2008), 1649–1663.CrossRefGoogle Scholar
11.Hou, J. C. and Di, Q. H., Maps preserving numerical range of operator products Proc. Am. Math. Soc. 134 (2006), 1435–1446.CrossRefGoogle Scholar
12.Jafarian, A. A. and Sourour, A. R., Spectrum-preserving linear maps J. Funct. Anal. 66 (1986), 255–261.CrossRefGoogle Scholar
13.Laursen, K. B. and Neumann, M. M., An introduction to local spectral theory, London Mathematical Society Monograph, New Series, vol. 20, 2000.CrossRefGoogle Scholar
14.Li, C. K., Šemrl, P. and Sze, N. S., Maps preserving the nilpotency of products of operators, Linear Algebra Appl. 424 (2007), 222–239.CrossRefGoogle Scholar
15.Molnár, L., Some characterizations of the automorphisms of B(H) and C(X) Proc. Am. Math. Soc. 130 (2002), 111–120.CrossRefGoogle Scholar
16.Omladič, M. and Šemrl, P., Additive mappings preserving operators of rank one, Linear Algebra Appl. 182 (1993), 239–256.CrossRefGoogle Scholar
17.Wang, M., Fang, L. and Ji, G., Linear maps preserving idempotency of products or triple Jordan products of operators, Linear Algebra Appl. 429 (2008), 181–189.CrossRefGoogle Scholar