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

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