Zero-dilation index of \(S_n\)-matrix and companion matrix (Q2834761)

The zero-dilation index \(d(A)\) of a complex \(n\times n\) matrix \(A\) is the largest integer \(k\) for which \(A\) is unitarily similar to a matrix with the \(k\times k\) zero submatrix in the left upper corner. The matrix \(A\) is an \(S_n\)-matrix if its spectral norm \(\|A\|\leq 1\), spectral radius \(\rho(A)<1\), and \(\text{rank}\,(I-A^*A)=1\). The authors prove that if \(A\) an \(S_n\)-matrix or a companion matrix, then \(d(A)\leq\lceil n/2\rceil\). They also give several equivalent conditions for equality.