Numerical ranges and compressions of \(S_{n}\)-matrices (Q2852263)

An \(n\times n\) complex matrix is called an \(S_n\)-matrix if \(A\) is a contraction with eigenvalues in the open unit disc and \(\text{rank}\, (I_n - A^*A) = 1\). Let \(W(A)\) be the classical numerical range of \(A\). This paper studies the properties of \(W(A)\) by relating it to its compression. The following results are established:NEWLINENEWLINE (1) If \(B\) is a \(k\times k\) (\(1\leq k<n\)) compression of \(A\), then \(W(B)\) is a proper subset of \(W(A)\),NEWLINENEWLINE (2) if \(A\) is in upper-triangular form and \(B\) is a \(k\times k\) (\(1\leq k <n\)) compression of \(A\), then the boundaries of \(W(A)\) and \(W(B)\) do not intersect, andNEWLINENEWLINE(3) \(2\) is the maximum value of \(k\) for which there is a \(k\times k\) compression of \(A\) with all its diagonal entries in the boundary of \(W(A)\) if \(n=2\); \([n/2]\) is the maximum value if \(n\geq 3\).