Preservers of local spectra of matrix sums
Let \(M_n(\mathbb{C})\) be the algebra of all \(n\times n\) complex matrices and \(e\) a nonzero vector of \(\mathbb{C}^n\). For a matrix \(T\in M_n(\mathbb{C})\), let \(\sigma_T(x)\) stand for the local spectrum of \(T\) at \(e\). Among other results established in [\textit{M. Bendaoud} et al., Linear Multilinear Algebra 61, No.~7, 871--880 (2013; Zbl 1278.47042)], it is shown that a surjective map \(\varphi\) on \(M_n(\mathbb{C})\) satisfies \(\varphi(0)=0\) and \[ \sigma_{\varphi(T)-\varphi(S)}(e)\subset\sigma_{T-S}(e)\;(T,S\in M_n(\mathbb{C})) \] if and only if there exists an invertible matrix \(A\in M_n(\mathbb{C})\) such that \(Ae=e\) and \(\varphi(T)=ATA^{-1}\) for all \(T\in M_n(\mathbb{C})\). The proof uses a density argument and the automatic continuity of \(\varphi\) that follows an argument quoted from [\textit{C. Costara}, Linear Algebra Appl. 435, No. 11, 2674--2680 (2011; Zbl 1231.15003)]. Furthermore, maps on \( M_n(\mathbb{C})\) verifying the reverse of the above inclusion are also characterized. In the paper under review, the author re-establishes similar results to the ones from the first cited paper, using sums of matrices instead of subtractions. The proofs are essentially the same.
scientific article; zbMATH DE number 6138430
scientific article