The Berger–Wang formula for the Markovian joint spectral radius

Abstract

We give a formula for the joint local spectral radius of a bounded subset of bounded linear operators on a Banach space X in terms of the dual of X. Let X be a Banach space and L(X) the algebra of all bounded linear operators in X. The joint spectral radius ρ(M) of a bounded subset M of L(X) was introduced by G.-C. Rota and W. G. Strang [5] as ρ(M) = lim sup n→∞ M n 1/n , where M n is the set of all products T 1 •.. .•T n (T i ∈ M) and M n = sup T ∈M n T. Recently the notion of the joint local spectral radius ρ x (M) at a point x ∈ X was introduced by R. Drnovšek [2] for a finite subset M of L(X) and by V. S. Shulman and Yu. V. Turovskii [6] for a bounded M ⊆ L(X) as ρ x (M) = lim sup n→∞

Matrix products with constraints on the sliding block relative frequencies of different factors

“…In this section we strengthen the results of [1,19] on the joint/generalized spectral radius for the Markovian matrix products for the matrix products (1) whose index sequences α = (α n ) are subjected to constraints on the sliding block relative frequencies of symbols.…”

“…A more complicated situation is when these matrix products are somehow constrained, for example, some combinations of matrices in them are forbidden. One of situations of the kind was investigated in [1,19], where the concepts of the Markovian joint and generalized spectral radii were introduced to analyze the matrix products with constraints of the Markovian type on the neighboring matrices. Another situation of the kind is described in what follows.…”

“…The problem on constructive determination of classes of symbolic sequences with prescribed frequency properties and of the related matrix products has got lesser attention. Only recently, in [1,19] the area of applicability of the methods of joint/generalized spectral radius was expanded on the matrix products (1) in which the index sequences α = (α n ) are described by the topological Markov chains. In connection with this, the aim of the article is to describe a class of r-symbolic sequences for which it is possible to constructively define "approximate relative frequencies" of symbols in the slidingblocks, that is, in each block of consecutive symbols.…”

“…In connection with this, the aim of the article is to describe a class of r-symbolic sequences for which it is possible to constructively define "approximate relative frequencies" of symbols in the slidingblocks, that is, in each block of consecutive symbols. Such a description will be given in terms of the -step topological Markov chains (subshifts of finite type) which allows to use the approach developed in [1,19] to define the joint/generalized spectral radius 2 for the products of matrices whose factors are applied not arbitrarily but are subjected to some frequency constraints. Outline the structure of the work.…”

“…Properties of symbolic sequences with constraints on the sliding block relative frequencies obtained in Section 2 are used in Section 3 to analyze the rate of growth of norms of matrix products. For this purpose, we recall the basics of the theory of joint/generalized spectral radius and the results from [1,19] related to the generalization of the Berger-Wang theorem on the case of Markovian products of matrices. In the final part of Section 3, we define the concepts of joint and generalized spectral radii for matrix products with constraints on the sliding -block relative frequencies, and deduce with the help of Theorem 1 from the results of [1,19] an analog of the Berger-Wang theorem for this case, Theorem 2.…”