A note on numerical range and product of matrices
Let \(A\) be a \(n\)-by-\(n\) matrix. We use \(\sigma(A)\) and \(W(A)\) for the spectrum and the numerical range of \(A\). A well-known result in matrix theory states that if \(A\) is positive semidefinite matrix, then \(\sigma(AB)\subseteq W(A)W(B)\) for all \(B\in M_n\). \textit{R. Alizadeh} [ibid. 437, No. 6, 1422--1425 (2012; Zbl 1259.15033)] proved the converse of this result. That is: Let \(A\in M_n\). Then \(\sigma(AB)\subseteq W(A)W(B)\) for all \(B\in M_n\) if and only if \(A\) is a scalar multiple of a positive semidefinite matrix. But if 0 is in the interior of \(W(A)\), then the presented proof does not work. In this paper, the authors prove this theorem again.
scientific article; zbMATH DE number 6141615