Zero-dilation indices and numerical ranges

The zero-dilation index $d(A)$ of a matrix $A$ is the largest integer $k$ for which $\begin{bmatrix}0_k& *\\ * & *\end{bmatrix}$ 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 $$\mathcal{C}=\begin{bmatrix} 0& \bigoplus_{j=1}^{m-1}A_j \\ B_0& [B_j]_{j=1}^{m-1}\end{bmatrix}\ \mbox{and}\ \mathcal{K}=\begin{bmatrix}0& A& A^2&\cdots& A^{m-1}\\ 0 & 0& A& \ddots& \vdots\\ 0& 0 &0 &\ddots& A^2\\ \vdots& \vdots &\vdots & \ddots& A\\ 0& 0 & 0& \cdots &0\end{bmatrix}$$ where $\mathcal{C}$ and $\mathcal{K}$ are $mn$-by-$mn$ and $A_j,B_j,A$ are $n$-by-$n$. Provided $\bigoplus_{j=1}^{m-1}A_j$ is nonsingular, it is proved that $d(\mathcal{C})$ satisfies the following: if $m\geq 3$ is odd (respectively, $m\geq 2$ is even), then $\frac{(m-1)n}{2}\leq d(\mathcal{C})\leq \frac{(m+1)n}{2}$ (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 $(\mathcal{K}+\mathcal{K}^*)/2$. Alternative characterizations of $d(\mathcal{K})$ are given. The circularity of the numerical range of $\mathcal{K}$ is also considered.