Preservers of eigenvalue inclusion sets of matrix products (Q609504)

Given a complex \(n\)-by-\(n\) matrix \(A\), there are several easy to calculate regions which are known to contain all its eigenvalues. The paper concentrates on the Ostrowski set \[ O_\varepsilon(A):=\bigcup_{k=1}^n \{\mu\in{\mathbb C};\;\;|\mu-a_{kk}|\leq \big(\sum_{j\neq k} |a_{kj}|\big)^\varepsilon \big(\sum\nolimits_{j\neq k} |a_{jk}|\big)^{1-\varepsilon}\};\qquad\varepsilon\in[0,1] \] and the Brouwer set \[ C(A):=\bigcup_{1\leq k<l\leq n}\{\mu\in{\mathbb C};\;\;|\mu-a_{kk}|\cdot |\mu-a_{ll}|\leq \big(\sum_{j\neq k} |a_{kj}|\big) \big(\sum_{j\neq l} |a_{kj}|\big)\}. \] It is then shown that a possibly nonlinear map \(\Phi\) is weakly multiplicative, in a sense that \(O_\varepsilon(\Phi(A)\Phi(B))=O_\varepsilon(AB)\) for every pair of matrices, if and only if \(\Phi\) is a signed or nonsigned similarity by a generalized permutation matrix, which moreover must be unitary for \(n\geq 3\). In particular, this implies automatic linearity and invertibility of \(\Phi\). A similar result holds in the case of Brouwer sets. The proofs start by establishing the image of a diagonal matrix diag\((1,\dots,n)\), and then proceeds with the images of matrix units. A similar result on weakly additive maps was obtained by \textit{J. Hartman}, \textit{A. Herman}, and \textit{C. K. Li} [Linear Algebra Appl. 433, No.~5, 1038--1051 (2010; Zbl 1232.15021)].