The k-rank numerical radii

TL;DR

This paper explores the properties of the k-rank numerical range and radii of matrices, expressing these ranges as intersections of numerical ranges related to isometries, thus advancing understanding of matrix spectral characteristics.

Contribution

It introduces a new representation of the k-rank numerical range using intersections of numerical ranges with respect to isometries, enhancing the theoretical framework of matrix analysis.

Findings

Expressed the k-rank numerical range as an intersection of numerical ranges.

Provided insights into the properties of k-rank numerical radii.

Extended the theoretical understanding of matrix spectral sets.

Abstract

The $k$-rank numerical range $\Lambda_{k}(A)$ is expressed via an intersection of a countable family of numerical ranges $\{F(M^{*}_{\nu}AM_{\nu})\}_{\nu\in\mathbb{N}}$ with respect to $n\times (n-k+1)$ isometries $M_{\nu}$. This implication for $\Lambda_{k}(A)$ provides further elaboration of the $k$-rank numerical radii of $A$.