Maps on matrices preserving local spectra

Abstract

For a square matrix T and a nonzero vector e ∈ ℂ ⁿ , let σ _T (x) be the local spectrum of T at e. Characterization is obtained for surjective maps φ on ℳ _n (ℂ) satisfying σ_{φ(T)−φ(S)}(e) ⊆ σ _T−S (e) for all matrices T and S. The same description is obtained by supposing that σ _T−S (e) ⊆ σ_{φ(T)−φ(S)}(e) for all matrices T and S, without the surjectivity assumption on φ. Continuous maps from ℳ _n (ℂ) onto itself that preserve the local spectral radius distance at a nonzero fixed vector are also characterized.

Keywords:

• local spectrum
• local spectral radius
• preserver problems

AMS Subject Classifications::

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

Acknowledgements

The authors thank Professor L. Molnár very much for his remarks and comments. They also thank the support and the hospitality of the organizers of the Coimbra meeting on Directions in Matrix Theory, University of Coimbra, Portugal, July 9–10, 2011, where part of the results in this article was presented by the first author.