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,$Sigma$, $r_*left(Sigmaright)= hat{r}left(Sigmaright)$, but for a bounded set of  upper triangular matrices with entries in a Banach algebra($Sigma$), $r_*left(Sigmaright)neqhat{r}left(Sigmaright)$. We  investigate when the set is  defective or not and equivalent properties for having a norm equal to jsr, too.