Linear maps preserving inclusion of fixed subsets into the spectrum

Abstract

Let n≥2 be a natural number, denote by Mn the space of all matrices of size n over an algebraically closed field F, and let K1,K2⊆F be two nonempty subsets, each having at most n elements. In the case when K2 has at least the number of elements of K1, we characterize linear bijective maps φ on Mn having the property that, for each n×n matrix T, we have that K2 is a subset of the spectrum of φ(T) whenever K1 is a subset of the spectrum of T. As an application, we characterize linear maps φ on Mn having the property that, for each n×n matrix T, we have that K1 is a subset of the spectrum of T if and only if K2 is a subset of the spectrum of φ(T), with no assumption on the cardinalities of K1 and K2.