Zero-dilation indices and numerical ranges

The zero-dilation index $d(A)$ of a matrix $A$ is the largest integer $k$ for which $\left[\begin{array}{cc}{0}_k& \ast \\ \ast & \ast \end{array}\right]$ 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 $$\mathcal{C}=\left[\begin{array}{cc}0& {⨁}_{j=1}^{m-1}{A}_j\\ B_0& [B_j{]}_{j=1}^{m-1}\end{array}\right]and\mathcal{K}=\left[\begin{array}{ccccc}0& A& A^2& \cdots & A^{m-1}\\ 0& 0& A& \ddots & \mathrm{⋮}\\ 0& 0& 0& \ddots & A^2\\ \mathrm{⋮}& \mathrm{⋮}& \mathrm{⋮}& \ddots & A\\ 0& 0& 0& \cdots & 0\end{array}\right]$$ where $\mathcal{C}$ and $\mathcal{K}$ 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 $d(\mathcal{C})$ satisfies the following: if $m\ge 3$ is odd (respectively, $m\ge 2$ is even), then $\frac{(m-1)n}{2}\le d(\mathcal{C})\le \frac{(m+1)n}{2}$ (respectively, $d(\mathcal{C})=\frac{mn}{2}$). 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, $d(\mathcal{K})$ is proved to be equal to the number of nonnegative eigenvalues of $(\mathcal{K}+{\mathcal{K}}^{\ast })\mathrm{/}2$. Alternative characterizations of $d(\mathcal{K})$ are given. The circularity of the numerical range of $\mathcal{K}$ is also considered.