Maps preserving the spectrum of polynomial products of matrices

[摘要] Let M-p (C) be the algebra of all p x p complex matrices, and let sigma(T) denote the spectrum of any matrix T is an element of M-p (C). For an integer n >= 2, let f(xi(1), ..., xi(n)) be a xi(1)-linear polynomial with complex coefficients and n-non commuting indeterminates xi(1), ..., xi(n) Under minor natural conditions on f, we show that if a map phi on M-p satisfies sigma(f (phi(T-1), ..., phi(T-n))) = sigma(f(T-1, ..., T-n)) (0.1) for all T-1, ..., T-n is an element of M-p (C), then there exist a nonzero scalar lambda and an invertible matrix A is an element of M-p (C) such that phi has one of the following forms: T bar right arrow lambda ATA(-1) for all T is an element of M-p (C), (0.2) or T bar right arrow lambda AT(t) A(-1) for all T is an element of M-p (C), (0.3) Here T-t denotes the transpose of T. In general, not all of mappings of the form (0.2) or the form (0.3) satisfy (0.1). When n = 2, we describe the set of all scalars lambda for which the mapping (0.2) or (0.3) satisfies (0.1). We also obtain analogue results of spectrum preserving maps on the real linear space H-p of all self-adjoint matrices in M-p (C). Our results extend and unify several results obtained earlier on maps on M-p (C) preserving the spectrum of generalized product or generalized Jordan product of matrices. (C) 2019 Elsevier Inc. All rights reserved.

[发布日期] 2019-12-15

[关键词] Nonlinear preservers;Matrices;Spectrum;Eigenvalue;Polynomials of non commuting variables