Zero-dilation indices and numerical ranges

TL;DR

This paper investigates the zero-dilation indices of specific block matrices, providing bounds, exact values, and characterizations, and explores their numerical ranges and eigenvalue relations.

Contribution

It introduces new bounds and exact formulas for the zero-dilation indices of block matrices like companion and KMS matrices, expanding understanding of their spectral properties.

Findings

For odd m, zero-dilation index ranges between ((m-1)n)/2 and ((m+1)n)/2.

For even m, zero-dilation index equals (mn)/2.

Zero-dilation index of KMS matrices equals the count of nonnegative eigenvalues of (K+K*)/2.

Abstract

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.