An operator—in general nonlinear—associated with a pair of nonnegative matrices, is defined and some of its spectral properties studied. If the pair of matrices are a square matrix A and the identity matrix of the same order, the operator reduces to the linear operator A. The results obtained include generalizations of one of the principal conclusions of the theorem of Perron-Frobenius.

We give a new proof of a result due to Ruelle about the existence and simplicity of a unique maximal eigenvalue for a Ruelle^Perron^Frobenius operator acting on some Ho« lder continuous function space. Mathematics Subject Classi¢cations (1991): Primary 58F23, Secondary 30C62. Key words: locally expanding, mixing, Ruelle^Perron^Frobenius operator, maximal eigenvalue. 1. Introduction Ruelle's The...

The Perron-Frobenius Theorem says that if A is a nonnegative square matrix some power of which is positive, ihen there exî sts an .v,, such thai A ".\/\\A "x\\ converges to x,j for all .v > 0, There are many classical proofs of this theorem, all depending on a connection between positivily of a matrix and properties of ils eigenvalues. A more modern proof, due to Garrett Birkhoff. is based on t...

We present a new extension of the well-known Perron-Frobenius theorem to regular matrix pairs (E, A). The new extension is based on projector chains and is motivated from the solution of positive differential-algebraic systems or descriptor systems. We present several examples where the new condition holds, whereas conditions in previous literature are not satisfied.

where {Fn, n ∈ N}, is the natural filtration associated to (Xn). We assume that X0 ∈ R d + and that random vectors ξn are such that for all n, Xn ∈ R d + almost surely. The Perron-Frobenius Theorem [10, pp. 3-4] states that M has a positive Perron root ρ. We call Xn “subcritical” if ρ < 1, “supercritical” if ρ > 1 and “critical” if ρ = 1. In the “subcritical” case, one has P(‖Xn‖ → ∞) = 0 becau...

This note presents a new proof of basic Perron{Frobenius theory of irreducible nonnegative matrices. ABSTRACT This note presents a new proof of basic Perron{Frobenius theory of irreducible nonnegative matrices.

The statistical properties of endomorphisms under the assumption that the associated Perron– Frobenius operator is quasicompact are considered. In particular, the central limit theorem, weak invariance principle and law of the iterated logarithm for sufficiently regular observations are examined. The approach clarifies the role of the usual assumptions of ergodicity, weak mixing, and exactness....