Diagonals and numerical ranges of weighted shift matrices
For an \(n\)-by-\(n\) matrix \(A\), the authors examine the maximum number \(k = k(A)\) for which there is a \(k\)-by-\(k\) compression of~\(A\) with all its diagonal entries in the boundary \(\partial W(A)\) of the numerical range \(W(A)\) of~\(A\). For any such compression, a~standard model under unitary equivalence for~\(A\) is given. This is then applied to determine the value of \(k(A)\) for \(A\) of size~\(3\) in terms of the shape of its numerical range \(W(A)\). A~case of special interest is when \(k(A)\) equals the size of~\(A\). For the special case of weighted shift matrices, a~result is obtained to exactly determine when \(k(A)\) equals~\(n\). This is so if and only if either \(| w_1|=\dotsc=| w_n|\) or \(n\) is even and \(| w_1|=| w_3|=\dotsc=| w_{n-1}|\) and \(| w_2|=| w_4|=\dotsc=| w_n|\). Also, a characterization of those matrices~\(A\) for which \(k(A)=n-1\) is put forward.
scientific article; zbMATH DE number 6127356
scientific article