Results on the generalized numerical ranges in max algebra

TL;DR

This paper introduces and studies the rank-$k$ numerical range in max algebra, along with related max joint numerical ranges, exploring their algebraic properties in the context of nonnegative matrices.

Contribution

It presents the first systematic study of rank-$k$ numerical ranges and related concepts in max algebra, extending classical numerical range theory.

Findings

Defined the max joint $k$-numerical range and max joint $C$-numerical range.

Established algebraic properties of these numerical ranges.

Provided foundational results for max algebra numerical range theory.

Abstract

Let $n$ and $k$ be two positive integers with $k\leq n$ and $C$ an $n \times n$ matrix with nonnegative entries. In this paper, the rank-$k$ numerical range in the max algebra setting is introduced and studied. The related notions of the max joint $k$-numerical range and the max joint $C$-numerical range of an entry-wise nonnegative matrix and an $m$-tuple of nonnegative matrices are also introduced. Some interesting algebraic properties of these concepts are investigated.