The numerical range of derivations
Let p, q, n be integers satisfying \(1\leq p\leq q\leq n\). The (p,q)- numerical range of an \(n\times n\) complex matrix A is defined by \(W_{p,q}(A)=\{E_ p((UAU^*)[q]):\) U unitary\(\}\), where for an \(n\times n\) complex matrix X, X[q] denotes its \(q\times q\) leading principal submatrix and \(E_ p(X[q])\) denotes the pth elementary symmetric function of the eigenvalues of X[q]. The authors prove that if \(2\leq p\leq q\leq n-2\), then \(W_{p,q}(A)\) is not convex for the normal matrix \(A=I_{n-2}\oplus (1+i)I_ 2.\) Combining with known results, this shows that \(W_{p,q}(A)\) is convex for all A if and only if \(p=1\) or \(q\geq n-1\). Then they study various inclusion relations and inequalities involving \(r_{p,q}(A)=\max \{| \mu |:\quad \mu \in W_{p,q}(A)\}.\) More detailed description is obtained when A belongs to various classes of matrices.
scientific article; zbMATH DE number 4107016
scientific article