Document Type: Research Paper

Authors

1 Department of Mathematics, Faculty of Science, Payame Noor University, P.O.Box 19395-3697, Tehran, Iran.

2 Department of Mathematics, University of Mohaghegh Ardabili, Ardabil, Iran.

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

[1] B.A. Barnes, The spectral theory of upper triangular matrices with entries in a Banach algebra, Math. Nachr., 241 (2002), pp. 5-20.

[2] M. Berger and Y. Wang, Bounded semigroups of matrices, Lin. Alg. Appl., 166 (1992), pp. 21-27.

[3] V. Blondel and Y. Nestero, Computationally efficient approximations of the joint spectral radius, SIAM J. Matrix Anal. Appl., 27.1 (2005), pp. 256-272.

[4] L. Elsner, The generalized spectral-radius theorem: an analytic-geometric proof, Lin. Alg. Appl., 220 (1995), pp. 151-159.

[5] L. Gurvits, Stability of discrete linear inclusion, Lin. Alg. Appl., 231 (1995), pp. 47-85.

[6] R.E. Harte, Spectral mapping theorems, Proc. Roy. lrish Acad. sect. A 72 (1972), pp. 89-107.

[7] R. Jungers, The joint spectral radius, Lecture Notes in Control and Information Sciences, vol. 385, Springer-Verlag, Berlin, 2009.

[8] H. Mohammadzadehkan, A. Ebadian, and K. Haghnejad Azar, Joint spectrum of n-tuple of upper triangular matrices with entries in a unitall Banach algebra, Math. Rep., 19 (2017), pp. 21-29.

[9] P. Rosenthal and A. Soltysiak, Formulas for the joint spectral radius of noncommutating Banach algebra elements, Proc. Amer. Math. Soc., 123 (1995), pp. 2705-2708.

[10] G. C. Rota and W. G. Strang, A note on the joint spectral radius, Indag. Math. 22 (1960), pp. 379-381.

[11] J. Theys, Joint spectral radius: theory and approximations, PhD Thesis, University of Louvain, (2005).