Linear maps preserving inclusion and equality of the spectrum to fixed sets (Q6586417)

Let \(n\geq 2\) be an integer and let \(\mathcal{M}_{n}\) denote the set of \(n\times n\) matrices over an algebraically closed field \(\mathbb{F}\) of characteristic zero. For \(T\in \mathcal{M}_{n}\) we denote by \(T^{t}\), \(\mathrm{tr}(T)\), and \(\sigma (T)\) its transpose, trace, and spectrum, respectively. The \(n\times n\) identity matrix is denoted by \(I_{n}\).\N\NMatrices having all their eigenvalues inside a fixed region \(D\) in the complex plane are usually called \(D\)-stable and sets of such matrices appear, as author pointed out, in, e.g., the theory of stability for dynamical systems. In [\textit{A. Guterman} et al., Linear Algebra Appl. 315, No. 1--3, 61--81 (2000; Zbl 0964.15004)] the authors proved the following result on maps preserving \(K\)-stable matrices.\N\NLet \(K\subseteq \mathbb{F}\) be a proper non-empty subset and \(\Phi \colon \mathcal{M}_{n}\rightarrow \mathcal{M}_{n}\) a bijective linear map such that \(\sigma (T)\subseteq K\) implies \(\sigma \left( \Phi (T)\right) \subseteq K\) for every \(T\in \mathcal{M}_{n}\).\N\N\begin{itemize}\N\item[(i)] If \(K=\{0\}\), there exist \(c\in \mathbb{F}\backslash \{0\}\) and \(U,V\in \mathcal{M}_{n}\) with \(U\) invertible such that either\N\[\N\Phi (T)=cUTU^{-1}+\left(\mathrm{tr}(T)\right) V\qquad (T\in \mathcal{M}_{n}), \tag{2}\N\]\Nor\N\[\N\Phi (T)=cUT^{t}U^{-1}+\left( \mathrm{tr}(T)\right) V\qquad (T\in \mathcal{M}_{n}). \tag{3}\N\]\N\N\item[(ii)] If \(K=\mathbb{F}\backslash \{0\}\), there exist invertible matrices \(U,V\in \mathcal{M}_{n}\) such that either\N\[\N\Phi (T)=UTV\qquad (T\in \mathcal{M}_{n}), \tag{4}\N\]\Nor\N\[\N\Phi (T)=UT^{t}V\qquad (T\in \mathcal{M}_{n}). \tag{5}\N\]\N\N\item[(iii)] If \(K\neq \{0\}\) and \(K\neq \mathbb{F}\backslash \{0\}\), there exist \(c,d\in \mathbb{F}\backslash \{0\}\) with \(c\neq 0\) and an invertible \(U\in \mathcal{M}_{n}\) such that either\N\[\N\Phi (T)=cUTU^{-1}+d\left( \mathrm{tr}(T)\right) I_{n}\qquad (T\in \mathcal{M}_{n}), \tag{6}\N\]\Nor\N\[\N\Phi (T)=cUT^{t}U^{-1}+d\left( \mathrm{tr}(T)\right) I_{n}\qquad (T\in \mathcal{M} _{n}). \tag{7}\N\]\N\N\end{itemize}\N\NIn the paper under review, the author proves two main results. In the first part of the paper, he studies a variant of the problem considered in the statement of the above theorem. He proves the following result.\N\NLet \(K_{1},K_{2}\subseteq \mathbb{F}\) be two non-empty proper subsets and let \(\Phi \colon \mathcal{M}_{n}\rightarrow \mathcal{M}_{n}\) be a linear map such that\N\[\N\sigma (T)\subseteq K_{1}\quad \text{if and only if}\quad \sigma \left( \Phi (T)\right) \subseteq K_{2}\quad \text{for every }T\in \mathcal{M}_{n}. \tag{\(*\)}\N\]\NThen there exists a nonzero scalar \(\alpha \) such that \(K_{2}=\alpha K_{1}\). Moreover:\N\N\begin{itemize}\N\item[(i)] If \(K_{1}=\{0\}\), then \(K_{2}=\{0\}\) as well, and there exist \(c\in \mathbb{F}\backslash \{0\}\) and \(U,V\in \mathcal{M}_{n}\) with \(U\) invertible and \(c+\mathrm{tr}(V)\neq 0\) such that \(\Phi \) is either of the form (2) or (3).\N\N\item[(ii)] If \(K_{1}=\mathbb{F}\backslash \{0\}\), then \(K_{2}=\mathbb{F} \backslash \{0\}\) as well, and there exist invertible matrices \(U,V\in \mathcal{M}_{n}\) such that \(\Phi \) is either of the form (4) or (5).\N\N\item[(iii)] If \(K_{1}\neq \{0\}\) and \(K_{1}\neq \mathbb{F}\backslash \{0\}\), there exist \(c,d\in \mathbb{F}\backslash \{0\}\) with \(c\neq 0\) and \(c+nd\neq 0\), and an invertible \(U\in \mathcal{M}_{n}\) such that \(K_{2}=(c+nd)K_{1}\) and \(\Phi \) is either of the form (6) or (7).\N\end{itemize}\N\NIn the second part of the paper, the author considers the same type of preserver problem, but in this case equalities are used in (\(*\)). Also, since an \(n\times n\) matrix has at most \(n\) distinct eigenvalues, some natural restrictions on the number of elements for the sets \(K_{1}\) and \(K_{2}\) are imposed. Namely, for \(K_{1},K_{2}\subseteq \mathbb{F}\) with \(1\leq \left\vert K_{j}\right\vert \leq n\) for \(j=1,2\), the author characterizes linear maps \(\Phi \colon \mathcal{M}_{n}\rightarrow \mathcal{M}_{n}\) such that\N\[\N\sigma (T)=K_{1}\quad \text{if and only if}\quad \sigma \left( \Phi (T)\right) =K_{2}\quad \text{for every }T\in \mathcal{M}_{n}.\N\]