Holomorphic maps preserving parts of the local spectrum
Let \(M_n\) denote the algebra of all \(n \times n\) complex matrices. It is well known that linear maps that preserve the spectrum or the spectral radius are of a standard form. This result has been extended in several directions. For example \textit{M. González} and \textit{M. Mbekhta} [Linear Algebra Appl. 427, No. 2--3, 176--182 (2007; Zbl 1127.15005)] have considered linear maps between matrix spaces that preserve the local spectrum. More recently, \textit{A. Bourhim} and \textit{V. G. Miller} [Stud. Math. 188, No. 1, 67--75 (2008; Zbl 1145.47005)] studied linear maps that preserve the local spectral radius at a point. In the present paper, the author extends these results to the following setting: fix a nonzero \(x_{0} \in M_{n}\), let \(U\) be a domain in \(M_n\) containing the zero matrix, and let \(\varphi : U \rightarrow M_n\) be a holomorphic map such that the local spectrum of \(T\) at \(x_0\) and the local spectrum of \(\varphi(T)\) at \(x_0\) have always a point in common, for all \(T \in U\). Then \(\varphi\) preserves the spectrum, i.e., \(T\) and \(\varphi (T)\) have the same spectrum for all \(T \in U\). It is also proved that a holomorphic map \(\varphi : U \rightarrow M_{n}\) with \(\varphi (0)=0\) that preserves the local spectral radius for every element of \(U\) at \(x_0\) satisfies that there exists a modulus one complex number \(\xi\) such that \(T\) and \(\xi\varphi(T)\) have the same spectrum for all \(T\) in \(U\). The author also gives some examples about the sharpness of the results.
scientific article; zbMATH DE number 6477368
scientific article