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Joint and generalized spectral radius of upper triangular matrices with entries in a unital Banach algebra. (English) Zbl 1474.46089

Summary: 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 as for 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.

MSC:

46H05 General theory of topological algebras
47A25 Spectral sets of linear operators
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