We extend a number of results from one dimensional dynamics based on spectral properties of the Ruelle-Perron-Frobenius transfer operator to Anosov diffeomorphisms on compact manifolds. This allows to develop a direct operator approach to study ergodic properties of these maps. In particular, we show that it is possible to define Banach spaces on which the transfer operator is quasicompact. (In...

A matrix A can be tested to determine whether it is eventually positive by ex1 amination of its Perron-Frobenius structure, i.e., by computing its eigenvalues and left and right 2 eigenvectors for the spectral radius ρ(A). No such “if and only if” test using Perron-Frobenius prop3 erties exists for eventually nonnegative matrices. The concept of a strongly eventually nonnegative 4 matrix was wa...

Given a discrete-time random dynamical system represented by cocycle of non-singular measurable maps, we may obtain information on quantities studying the Perron–Frobenius operators associated to maps. Of particular interest is second-largest Lyapunov exponent for operators, λ2, which can tell us about mixing rates and decay correlations in system. We prove generalized theorem cocycles bounded ...

The main results of this paper are generalizations of classical results from the numerical range to the block numerical range. A different and simpler proof for the Perron-Frobenius theory on the block numerical range of an irreducible nonnegative matrix is given. In addition, the Wielandt's lemma and the Ky Fan's theorem on the block numerical range are extended.

We present a new approach of proving certain Carath\'{e}odory-type theorems using the Perron-Frobenius Theorem, classical result in matrix theory describing largest eigenvalue with positive entries. One problems left open this note is whether our may be extended to prove similar results area, particular Colourful Carath\'{e}odory Theorem.

we give further results for perron-frobenius theory on the numericalrange of real matrices and some other results generalized from nonnegative matricesto real matrices. we indicate two techniques for establishing the main theorem ofperron and frobenius on the numerical range. in the rst method, we use acorresponding version of wielandt's lemma. the second technique involves graphtheory.

Motivated by the classical results of G. Frobenius and O. Perron on the spectral theory of square matrices with nonnegative real entries, D. Evans and R. Høegh-Krohn have studied the spectra of positive linear maps on general (noncommutative) matrix algebras. The notion of irreducibility for positive maps is required for the Frobenius theory of positive maps. In the present article, irreducible...