Joint and Generalized Spectral Radius of Upper Triangular Matrices with Entries in a Unital Banach Algebra

Abstract

In this paper, we discuss some properties of joint spectral {radius(jsr)} and generalized spectral radius(gsr) for a finite set of upper triangular matrices with entries in a Banach algebra and represent relation between geometric and joint/generalized spectral radius. Some of these are in scalar matrices, but some are different. For example for a bounded set of scalar matrices,$\Sigma$, $r_*\left(\Sigma\right)= \hat{r}\left(\Sigma\right)$, but for a bounded set of upper triangular matrices with entries in a Banach algebra($\Sigma$), $r_*\left(\Sigma\right)\neq\hat{r}\left(\Sigma\right)$. We investigate when the set is defective or not and equivalent properties for having a norm equal to jsr, too.

Keywords

• banach algebra
• upper triangular matrix
• generalized spectral radius
• joint spectral radius
• geometric joint spectral radius