Finiteness Property of a Bounded Set of Matrices with Uniformly Sub-Peripheral Spectrum

TL;DR

This paper presents a simple condition under which a bounded set of matrices exhibits the finiteness property, based on the spectral behavior of matrix products with uniformly sub-peripheral spectra.

Contribution

It introduces a new criterion linking the spectral properties of matrix products to the finiteness property of the set, expanding understanding of spectral radius behavior.

Findings

Existence of a sequence with spectra tending to the joint spectral radius guarantees finiteness.

Uniformly sub-peripheral spectra are key to establishing the finiteness property.

Provides a practical condition for verifying the spectral finiteness property.

Abstract

In the paper, a simple condition guaranteing the finiteness property for a bounded set of matrices is presented. Given a bounded set S of real or complex matrices, it is shown that existence of a sequence of matrix products such that the spectrum of each matrix in this sequence is uniformly sub-peripheral and tends to the joint spectral radius of S, guarantees the spectral finiteness property for S.