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

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