Related papers: Towards a Perron--Frobenius Theory for Eventually …

We develop a systematic theory of eventually positive semigroups of linear operators mainly on spaces of continuous functions. By eventually positive we mean that for every positive initial condition the solution to the corresponding Cauchy…

Let $T$ denote a positive operator with spectral radius $1$ on, say, an $L^p$-space. A classical result in infinite dimensional Perron--Frobenius theory says that, if $T$ is irreducible and power bounded, then its peripheral point spectrum…

The classical Perron-Frobenius theory asserts that an irreducible matrix $A$ has cyclic peripheral spectrum and its spectral radius $r(A)$ is an eigenvalue corresponding to a positive eigenvector. In Radjavi (1999) and Radjavi and Rosenthal…

The spectral theory of semigroup generators is a crucial tool for analysing the asymptotic properties of operator semigroups. Typically, Tauberian theorems, such as the ABLV theorem, demand extensive information about the spectrum to derive…

This paper considers strongly continuous semigroups of operators on Banach lattices which are locally eventually positive, a property that was first investigated in the context of concrete fourth-order evolution equations. We construct a…

We work in an ordered Banach space with closed generating positive cone. We show that a positive compact operator has zero spectral radius or a positive eigenvector with the corresponding eigenvalue equal to the spectral radius.

This paper contributes to the analysis of the peripheral (point) spectrum of positive linear operators on Banach lattices. We show that, under appropriate growth and regularity conditions, the peripheral point spectrum of a positive…

We perform an in-depth study of some domination and smoothing properties of linear operators and of their role within the theory of eventually positive operator semigroups. On the one hand we prove that, on many important function spaces,…

An intriguing feature of positive $C_0$-semigroups on function spaces (or more generally on Banach lattices) is that their long-time behaviour is much easier to describe than it is for general semigroups. In particular, the convergence of…

We initiate a theory of locally eventually positive operator semigroups on Banach lattices. Intuitively this means: given a positive initial datum, the solution of the corresponding Cauchy problem becomes (and stays) positive in a part of…

Let $T$ be a positive operator on a complex Banach lattice. It is a long open problem whether the peripheral spectrum $\sigma_{\operatorname{per}}(T)$ of $T$ is always cyclic. We consider several growth conditions on $T$, involving its…

The spectral properties of the Frobenius-Perron operator of one-dimensional maps are studied when approaching a weakly intermittent situation. Numerical investigation of a particular family of maps shows that the spectrum becomes extremely…

The notion \emph{Perron-Frobenius theory} usually refers to the interaction between three properties of operator semigroups: positivity, spectrum and long-time behaviour. These interactions gives rise to a profound theory with plenty of…

We consider eventually positive operator semigroups and study the question whether their eventual positivity is preserved by bounded perturbations of the generator or not. We demonstrate that eventual positivity is not stable with respect…

The classic result of Perron and Frobenius states that if $A$ and $B$ are matrices with nonnegative elements, such that $A \leq B$, $A$ is irreducible, and $\rho(A) = \rho(B)$ then $A = B$. We extend this result to a large class of band…

We characterize the spectrum of positive linear operators $T:X \to Y$, where $X$ and $Y$ are complex Banach function spaces with unit $1$, having finite rank and a partition of unity property. Then all the points in the spectrum are…

In this review paper we carry on our investigations on Schroedinger operators with inverse square potentials on the half-line. Depending on several parameters, such operators possess either a finite number of complex eigenvalues, or an…

The Perron--Frobenius theorem in infinite-dimensional Hilbert spaces can be breifly stated as follows: Given a Hilbert cone in a real Hilbert space, a bounded positive self-adjoint operator $A$ is ergodic with respect to this cone if and…

We study the spectrum of one dimensional integral operators in bounded real intervals of length $2L$, for value of $L$ large. The integral operators are obtained by linearizing a non local evolution equation for a non conserved order…

We consider linear operators defined on a subspace of a complex Banach space into its topological antidual acting positively in a natural sense. The goal of this paper is to investigate of this kind of operators. The main theorem is a…