A Simple Approach to the Perron-Frobenius Theory for Positive Operators on General Partially-Ordered Finite-Dimensional Linear Spaces41

Compact weighted Frobenius-Perron operators and their spectra

In this note we characterize the compact weighted Frobenius-Perron operator $p$ on $L^1(Sigma)$ and determine their spectra. We also show that every weakly compact weighted Frobenius-Perron operator on $L^1(Sigma)$ is compact.

Spectrum Properties of the Koopman and Frobenius-Perron Operators

Positive and Z-operators on closed convex cones

Let K be a closed convex cone with dual K∗ in a finite-dimensional real Hilbert space V . A positive operator on K is a linear operator L on V such that L (K) ⊆ K. Positive operators generalize the nonnegative matrices and are essential to the Perron-Frobenius theory. We say that L is a Z-operator on K if 〈L (x), s〉 ≤ 0 for all (x, s) ∈ K ×K such that 〈x, s〉 = 0. The Z-operators are generalizat...

PERRON-FROBENIUS THEORY ON THE NUMERICAL RANGE FOR SOME CLASSES OF REAL MATRICES

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.

Some results on the block numerical range

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.