Preservers of local spectra of operator products

Abstract

We characterize mappings on , the set of complex matrices, which preserve the local spectrum of the product of matrices at a nonzero fixed vector. Mappings from the algebra of all bounded linear operators on a Banach space into itself that preserve the local spectrum or the local spectral radius zero of the product of operators at all vectors are also classified.

Keywords:

• local spectrum
• local spectral radius
• nonlinear preservers

AMS Subject Classifications:

• Primary: 47B49
• Secondary: 47A10
• 47A53

Acknowledgements

The authors thank very much Professor L. Molnár for fruitful discussions and useful comments. They also wish to express their thanks to the anonymous referees for their comments and remarks. The research of the first and the third authors was partially supported by grants from ARS-MIU projects and the Moulay Ismail University Scientific Research Actions.