Extremal Lp-norms of linear operators and self-similar functions

AbstractWe prove that for any p∈[1,+∞] a finite irreducible family of linear operators possesses an extremal norm corresponding to the p-radius of these operators. As a corollary, we derive a criterion for the Lp-contractibility property of linear operators and estimate the asymptotic growth of orbits for any point. These results are applied to the study of functional difference equations with linear contractions of the argument (self-similarity equations). We obtain a sharp criterion for the existence and uniqueness of solutions in various functional spaces, compute the exponents of regularity, and estimate moduli of continuity. This, in particular, gives a geometric interpretation of the p-radius in terms of spectral radii of certain operators in the space Lp[0,1]

A relaxation scheme for computation of the joint spectral radius of matrix sets

The problem of computation of the joint (generalized) spectral radius of matrix sets has been discussed in a number of publications. In the paper an iteration procedure is considered that allows to build numerically Barabanov norms for the irreducible matrix sets and simultaneously to compute the joint spectral radius of these sets.Comment: 16 pages, 2 figures, corrected typos, accepted for publication in JDE

Continuity of the von Neumann entropy

A general method for proving continuity of the von Neumann entropy on subsets of positive trace-class operators is considered. This makes it possible to re-derive the known conditions for continuity of the entropy in more general forms and to obtain several new conditions. The method is based on a particular approximation of the von Neumann entropy by an increasing sequence of concave continuous unitary invariant functions defined using decompositions into finite rank operators. The existence of this approximation is a corollary of a general property of the set of quantum states as a convex topological space called the strong stability property. This is considered in the first part of the paper.Comment: 42 pages, the minor changes have been made, the new applications of the continuity condition have been added. To appear in Commun. Math. Phy

Computing the growth of the number of overlap-free words with spectra of matrices

Overlap-free words are words over the alphabet A = {a, b} that do not contain factors of the form xvxvx, where x isin A and v isin A*. We analyze the asymptotic growth of the number u/sub n/ of overlap-free words of length n. We obtain explicit formulas for the minimal and maximal rates of growth of u/sub n/ in terms of spectral characteristics (the lower spectral radius and the joint spectral radius) of one set of matrices of dimension 20. Using these descriptions we provide estimates of the rates of growth that are within 0.4% and 0.03% of their exact value. The best previously known bounds were within 11% and 3% respectively. We prove that u/sub n/ actually has the same growth for "almost all" n. This "average" growth is distinct from the maximal and minimal rates and can also be expressed in terms of a spectral quantity (the Lyapunov exponent). We use this expression to estimate it.Anglai