Complex Analysis and Operator Theory

, Volume 12, Issue 4, pp 859–868 | Cite as

Non-surjective Spectral Isometries on Matrix Spaces

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Abstract

We prove that if \(\varphi \,{:}\, {\mathcal {M}}_{m}\rightarrow {\mathcal {M}}_{n}\) is a linear map such that the spectral radius of \(x \in {\mathcal {M}}_{m}\) equals the spectral radius of \(\varphi (x) \in {\mathcal {M}}_{n}\) for each \(x \in {\mathcal {M}}_{m}\), there exists then a unimodular constant \(\xi \) such that the spectrum of \(\varphi (x) \) in \({\mathcal {M}}_{n}\) contains the spectrum of \(\xi x \in {\mathcal {M}}_{m}\) for each x. Structural informations on the map \(\varphi \) in a particular case are also obtained.

Keywords

Matrix spaces Spectral isometries Spectrum-preserving maps 

Mathematics Subject Classification

47B49 47A10 47A65 

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Faculty of Mathematics and InformaticsOvidius University of ConstanţaConstanţaRomania

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