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