Weakening the conditions in some classical theorems on linear preserver problems

Abstract

Linear preserver problems concern the study of linear maps on matrices or operators preserving special properties. In all previous results, one needs to know certain information of the maps on an infinite number of matrices to characterize the structures of a class of linear preservers. We prove that if a linear map preserves the determinant or eigenvalues of a finite number of complex matrices of a given order , then it preserves the corresponding property of all complex matrices of order . These results weaken the conditions in some classical theorems on linear preserver problems. We also present an application of linear preservers in quantum information science.

Keywords:

• determinant
• eigenvalue
• linear preserver

AMS Subject Classifications:

• 15A04
• 15A15
• 15A18

Acknowledgements

The author is grateful to Professor Chi-Kwong Li and Professor Xingzhi Zhan for helpful discussions and suggestions.

Notes

No potential conflict of interest was reported by the author.

Additional information

Funding

This research was supported by the NSFC grant 11401197 and a Fundamental Research Fund for the Central Universities.