Zero-dilation indices and numerical ranges

The zero-dilation index $d\left(A\right)$ of a matrix A is the largest integer k for which $\left[\begin{array}{cc}{0}_k& ⁎\\ ⁎& ⁎\end{array}\right]$ is unitarily similar to A. In this study, the zero-dilation indices of certain block matrices are considered, namely, the block matrix analogues of companion matrices and upper triangular KMS matrices, respectively shown as$C=\left[\begin{array}{cc}0& {⨁}_{j=1}^{m-1}{A}_j\\ B_0& {\left[B_j\right]}_{j=1}^{m-1}\end{array}\right]\text{and}K=\left[\begin{array}{ccccc}0& A& A^2& \cdots & A^{m-1}\\ 0& 0& A& \ddots & ⋮\\ 0& 0& 0& \ddots & A^2\\ ⋮& ⋮& ⋮& \ddots & A\\ 0& 0& 0& \cdots & 0\end{array}\right]$ where $C$ and $K$ are mn-by-mn and $A_j,B_j,A$ are n-by-n. Provided ${⨁}_{j=1}^{m-1}{A}_j$ is nonsingular, it is proved that $d\left(C\right)$ satisfies the following: if $m\ge 3$ is odd (respectively, $m\ge 2$ is even), then $\frac{(m-1)n}{2}\le d\left(C\right)\le \frac{(m+1)n}{2}$ (respectively, $d\left(C\right)=\frac{mn}{2}$). In the odd m case, examples are given showing that it is possible to get as zero-dilation index each integer value between $\frac{(m-1)n}{2}$ and $\frac{(m+1)n}{2}$. On the other hand, $d\left(K\right)$ is proved to be equal to the number of nonnegative eigenvalues of $(K+K^{⁎})/2$. Alternative characterizations of $d\left(K\right)$ are given. The circularity of the numerical range of $K$ is also considered.