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→∞
In the paper, it is proposed one more proof of a counterexample of the Finiteness Conjecture fulfilled in a traditional manner of the theory of dynamical systems. It is presented description of the structure of trajectories with the maximal growing rate in terms of extremal norms and associated with them so-called extremal trajectories. The construction of the counterexample is based on a detailed analysis of properties of extremal norms of two-dimensional positive matrices in which the technique of the Gram symbols is essentially used. At last, notions and properties of the rotation number for discontinuous orientation preserving circle maps play significant role in the proof.
Recently Blondel, Nesterov and Protasov proved [1,2] that the finiteness conjecture holds for the generalized and the lower spectral radii of the sets of non-negative matrices with independent row/column uncertainty. We show that this result can be obtained as a simple consequence of the so-called hourglass alternative used in [3], by the author and his companions, to analyze the minimax relations between the spectral radii of matrix products. Axiomatization of the statements that constitute the hourglass alternative makes it possible to define a new class of sets of positive matrices having the finiteness property, which includes the sets of non-negative matrices with independent row uncertainty. This class of matrices, supplemented by the zero and identity matrices, forms a semiring with the Minkowski operations of addition and multiplication of matrix sets, which gives means to construct new sets of non-negative matrices possessing the finiteness property for the generalized and the lower spectral radii.
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