Numerical ranges and Geršgorin discs (Q1931728)

Let \(A=[a_{ij}]\) be an \(n\times n\) complex matrix and \(W(A)\) be its numerical range. Define the Geršgorin region \(G(A)\) to be the convex hull of \(\cup_{i=1}^n\{z \in \mathbb{C}: |z-a_{ii}| \leq \left(\sum_{i\neq j}(|a_{ij}|+|a_{ji}|)\right)/2\}\) and the unitarily reduced Geršgorin region \(G'(A)\) to be \(\cap\{G(U^*AU): U \text{ is an } n\times n \text{ unitary matrix}\}\). It is known that \(W(A)\) is contained in \(G(A)\) and hence in \(G'(A)\), see \textit{C. Johnson} [Proc. Am. Math. Soc. 41, 57--60 (1973; Zbl 0248.15014)]. In the paper under review the authors present a decomposition theorem and use it to discuss conditions for \(W(A)\) to be equal to \(G(A)\) or \(G'(A)\). They show that if \(W(A) = G'(A)\), then the boundary of \(W(A)\) consists only of circular arcs and line segments. If, moreover, \(A\) is unitarily irreducible, then \(W(A)\) is a circular disc. They also give criteria for the equality of \(W(A)\) and \(G(A)\). In particular, it is shown that such \(A\)'s among the permutationally irreducible ones must have even sizes. They also characterize those \(A\)'s with size \(2\) or \(4\) satisfying \(W(A) = G(A)\) by showing that this occurs if and only if \(A\) is permutationally similar to a certain direct sum.