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Let m, n, p and q be positive integers such that p q ≤ m n and min (p , q) ≤ min (m , n). Let M m , n (C) denote the vector space of matrices of m rows and n columns with complex entries and let M n (C) stand for M n , n (C). For any matrix A ∈ M m (C) let σ (A) denote its spectrum. We prove that a couple of maps ϕ : M m , n (C) → M p , q (C) a n d ψ : M n , m (C) → M q , p (C) satisfy σ ϕ (M) ψ (N) = σ (M N) for all (M , N) ∈ M m , n (C) × M n , m (C) , if and only if there exist invertible matrices P ∈ M p (C) and Q ∈ M q (C) such that one of the following assertions holds: (m , n) = (p , q) , and we have ϕ (M) = P M Q and ψ (N) = Q − 1 N P − 1 for all (M , N) ∈ M m , n (C) × M n , m (C). (m , n) = (q , p) , and we have ϕ (M) = P M t Q and ψ (N) = Q − 1 N t P − 1 for all (M , N) ∈ M m , n (C) × M n , m (C). [ABSTRACT FROM AUTHOR]

Let be the algebra of all bounded linear operators on an infinite-dimensional complex Hilbert space , and let and be two subsets of containing all rank one operators. Fix , and let and denote the ϵ-pseudo spectrum and the ϵ-pseudo spectral radius of any operator. We show that if a surjective map satisfies then either there exists a unitary operator U on such that for all , or there is a conjugate unitary operator U on such that for all. We also characterize surjective maps satisfying [ABSTRACT FROM AUTHOR]

Let be the algebra of all bounded linear operators on a complex Banach space X. For an operator , let be the local spectral radius of T at any vector. For an integer , let be a finite sequence such that and at least one of the terms in appears exactly once. The generalized product of k operators is defined by and includes the usual product TS and the triple product TST. We show that a surjective map ϕ on satisfies for all and all if and only if there exists a map such that for all. [ABSTRACT FROM AUTHOR]