The numerical range of products of normal matrices
Let \(f_i,g_i\), \(1\leq i\leq n\) be complex numbers, \(\Sigma_n\) the simplex of nonnegative \(n\)-tuples summing to unity. Define for \(t\in \Sigma_n\) the functions \(f(t)=\sum_j t_i f_i\), \(\varphi (t)=\sqrt{\sum_{i<j} t_i t_j |f_i-f_j|^2}.\) Define \(g(s)\) and \(\psi(s)\) similarly for \(s\in \Sigma_n\) and the \(g_i.\) For \(X,Y\subseteq \{1,\ldots,n\}\) let \({\mathcal C}(X,Y):=\{z\in \mathbb C: |z-f(t)g(s)|\leq \varphi(t) \psi(s)\); \(\text{supp}(t)\subseteq X\), \(\text{supp}(s)\subseteq Y \}.\) (The support of an n-tuple is the set of indices on which it is \(\neq 0.\)) Theorem: The union of the sets \({\mathcal C}(X,Y)\) with \(|X|,|Y|\leq 3\) coincides with the union of numerical ranges \(W(FG)\) where \(F,G\) run over the normal \(n\times n\) matrices \(F,G\) with eigenvalues \(f_1,\ldots f_n; g_1,\ldots, g_n.\) The proof rests on a formula for the trace class multiplier norm of a matrix of rank \(\leq 2\) derived by \textit{S. W. Drury} [Linear Algebra Appl. 280, No. 2-3, 217-227 (1998; Zbl 0935.15023)] and on use of von Neumann's minimax theorem. In an earlier section the author uses the results of that paper also to supplement his alternative approach to the maximal spectral distance problem solved by \textit{J. Holbrook, M. Omladic}, and \textit{P. Šemrl} [ibid. 249, 197-205 (1996; Zbl 0861.15019)] by a geometric description for the 2-dimensional case.
scientific article; zbMATH DE number 1331350
scientific article