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Linear Operators Preserving Similarity Classes and Related Results

Published online by Cambridge University Press:  20 November 2018

Chi-Kwong Li  and
Stephen Pierce
Chi-Kwong Li
Affiliation:
Department of Mathematics, The College of William and Mary Williamsburg, Virginia 23187 U.S.A.
Stephen Pierce
Affiliation:
Department of Mathematical Sciences, San Diego State University, San Diego, California 92182, U.S.A.
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Abstract

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Let Mn be the algebra of n × n matrices over an algebraically closed field of characteristic zero. For A ∊ Mn, denote by the collection of all matrices in Mn that are similar to A. In this paper we characterize those invertible linear operators ϕ on Mn that satisfy , where for some given A1,..., Ak ∊ Mn and denotes the (Zariski) closure of S. Our theorem covers a result of Howard on linear operators mapping the set of matrices annihilated by a given polynomial into itself, and extends a result of Chan and Lim on linear operators commuting with the function f(x) = xk for a given positive integer k ≥ 2. The possibility of weakening the invertibility assumption in our theorem is considered, a partial answer to a conjecture of Howard is given, and some extensions of our result to arbitrary fields are discussed.

Keywords

15A42
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 37 , Issue 3 , 01 September 1994 , pp. 374 - 383
Copyright
Copyright © Canadian Mathematical Society 1994

References

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