Linear maps preserving the inclusion of fixed subsets into the local spectrum at some fixed vector

Abstract

For a natural number $$n \ge 2$$ n ≥ 2 , denote by $$\mathcal {M}_{n}$$ M n the space of all $$n\times n$$ n × n matrices over the complex field $$\mathbb {C}$$ C . Let $$x_0 \in \mathbb {C}^{n}$$ x 0 ∈ C n be a fixed nonzero vector, and fix also two nonempty subsets $$K_1, K_2 \subseteq \mathbb {C}$$ K 1 , K 2 ⊆ C , each having at most n distinct elements. Under the assumption that $$|K_1| \le |K_2|$$ | K 1 | ≤ | K 2 | , we characterize linear bijective maps $$\varphi $$ φ on $$\mathcal {M}_{n}$$ M n having the property that, for each matrix T, we have that $$K_2$$ K 2 is a subset of the local spectrum of $$\varphi (T)$$ φ ( T ) at $$x_0 $$ x 0 whenever $$K_1 $$ K 1 is a subset of the local spectrum of T at $$x_0$$ x 0 . As a corollary, we also characterize linear maps $$\varphi $$ φ on $$\mathcal {M} _{n}$$ M n having the property that, for each matrix T, we have that $$K_1$$ K 1 is a subset of the local spectrum of T at $$x_0$$ x 0 if and only if $$K_2$$ K 2 is a subset of the local spectrum of $$\varphi (T)$$ φ ( T ) at $$x_0$$ x 0 , without the bijectivity assumption on the map $$\varphi $$ φ and with no assumption made regarding the number of elements of $$K_1$$ K 1 and $$K_2$$ K 2 .