Maps preserving the spectrum of polynomial products of matrices.

Let M p (C) be the algebra of all p × p complex matrices, and let σ (T) denote the spectrum of any matrix T ∈ M p (C). For an integer n ≥ 2 , let f (ξ 1 , ... , ξ n) be a ξ 1 -linear polynomial with complex coefficients and n -non commuting indeterminates ξ 1 , ... , ξ n. Under minor natural conditions on f , we show that if a map φ on M p (C) satisfies (0.1) σ (f (φ (T 1) ,... , φ (T n))) = σ (f (T 1 ,... , T n)) for all T 1 ,... , T n ∈ M p (C) , then there exist a nonzero scalar λ and an invertible matrix A ∈ M p (C) such that φ has one of the following forms: (0.2) T ↦ λ A T A − 1 for all T ∈ M p (C) , or (0.3) T ↦ λ A T t A − 1 for all T ∈ M p (C). 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 λ 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. [ABSTRACT FROM AUTHOR]