Asymptotic stability and generalized Gelfand spectral radius formula

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...

On accuracy of approximation of the spectral radius by the Gelfand formula

The famous Gelfand formula ρ(A) = lim supn→∞ ‖A ‖ for the spectral radius of a matrix is of great importance in various mathematical constructions. Unfortunately, the range of applicability of this formula is substantially restricted by a lack of estimates for the rate of convergence of the quantities ‖A‖ to ρ(A). In the paper this deficiency is made up to some extent. By using the Bochi inequa...

A formula for the inner spectral radius

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...