A generalized spectral radius formula and Olsenʼs question

A formula for the inner spectral radius

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

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_*...

On explicit a priori estimates of the joint spectral radius by the generalized Gelfand formula

In various problems of control theory, non-autonomous and multivalued dynamical systems, wavelet theory and other fields of mathematics information about the rate of growth of matrix products with factors taken from some matrix set plays a key role. One of the most prominent quantities characterizing the exponential rate of growth of matrix products is the so-called joint or generalized spectra...

The generalized spectral radius and extremal norms

The generalized spectral radius, also known under the name of joint spectral radius, or (after taking logarithms) maximal Lyapunov exponent of a discrete inclusion is examined. We present a new proof for a result of Barabanov, which states that for irreducible sets of matrices an extremal norm always exists. This approach lends itself easily to the analysis of further properties of the generali...

The generalized Berger-Wang formula and the spectral radius of linear cocycles

Using multiplicative ergodic theory we prove two formulae describing the relationships between different joint spectral radii for sets of bounded linear operators acting on a Banach space. In particular we recover a formula recently proved by V. S. Shulman and Yu. V. Turovskĭı using operatortheoretic ideas. As a byproduct of our method we answer a question of J. E. Cohen on the limiting behavio...