Perturbations of positive semigroups and applications to population genetics (original) (raw)

A folk theorem in the spectral theory of C 0 -semigroups

Pacific Journal of Mathematics, 1984

If A is the infinitesimal generator of a C 0-semigroup T(t\ a classical theorem of Hille and Phillips relates the point spectrum of A and that of F(£) for £ > 0. Specifically, if μ is in the point spectrum of Γ(£) and μ Φ 0, then there exists <x 0 in the point spectrum of A with exp(ξα 0) = μ and the null space of μ-T(ξ) is the closed linear span of the null spaces of a n-A for a n = a 0 4-2πinξ~] and n ranging over the integers. In this note we shall extend the Hille-Phillips theorem by proving that the null space of (μ-T(ξ)) k is the closed linear span of the null spaces of (a n-A) k as n ranges over the integers. Such a result is useful in relating the order of poles of the resolvent of A and the order of poles of the resolvent of Γ(£), and as an example we shall give an application to the theory of positive (in the sense of cone-preserving) linear operators.

Boundedness properties of resolvents and semigroups of operators

Banach Center Publications, 1997

T j x 2 ≤ M (T) 2 x 2 is satisfied. Also suppose that the adjoint T * of the operator T is square bounded in average with constant M (T *). Then the operator T is power bounded in the sense that sup{ T n : n ∈ N} is finite. In fact the following inequality is valid for all n ∈ N: T n ≤ eM (T)M (T *). Suppose that T has a bounded everywhere defined inverse S with the property that for λ in the open unit disc of C the operator (I − λS) −1 exists and that the expression sup{(1 − |λ|) (I − λS) −1 : |λ| < 1} is finite. If T is power bounded, then so is S and hence in such a situation the operator T is similar to a unitary operator. If both the operators T * and S are square bounded in average, then again the operator T is similar to a unitary operator. Similar results hold for strongly 1991 Mathematics Subject Classification: 47A30, 47D05. Key words and phrases: power bounded operator, bounded semigroup, operator Poisson kernel, square bounded in average. The author is grateful to a number of people who in one way or another were involved during the preparation of this paper: F. Delbaen, L. Waelbroeck (Brussels), J. Zemánek (Warsaw). The author also wants to thank the National Fund for Scientific Research (NFWO) and the University of Antwerp (UIA) for their material support. The author is indebted to the referee for pointing out some errors in an earlier draft of the paper. Finally, the author is obliged to the "International Scientific and Technical Cooperation BLEU-Poland" for making it possible to visit the Banach Center in Warsaw in April 1994. The paper is in final form and no version of it will be published elsewhere. [59] 60 J. A. VAN CASTEREN continuous semigroups instead of (powers) of a single operator. Some results are also given in the more general Banach space context.

Local spectra and individual stability of uniformly bounded 𝐶₀-semigroups

We study the asymptotic behaviour of individual orbits T (·)x of a uniformly bounded C 0 -semigroup {T (t)} t≥0 with generator A in terms of the singularities of the local resolvent (λ − A) −1 x on the imaginary axis. Among other things we prove individual versions of the Arendt-Batty-Lyubich-Vũ theorem and the Katznelson-Tzafriri theorem.

The eigenstructure of some positive linear operators

2014

Of concern is the study of the eigenstructure of some classes of positive linear operators satisfying particular conditions. As a consequence, some results concerning the asymptotic behaviour as \(t\to +\infty\) of particular strongly continuous semigroups \((T(t))_{t\geq 0}\) expressed in terms of iterates of the operators under consideration are obtained as well. All the analysis carried out herein turns out to be quite general and includes some applications to concrete cases of interest, related to the classical Beta, Stancu and Bernstein operators.

Approximation by Polynomials with Nonnegative Coefficients and the Spectral Theory of Positive Operators

1998

For Σ a compact subset of C symmetric with respect to conjugation and f : Σ → C a continuous function, we obtain sharp conditions on f and Σ that insure that f can be approximated uniformly on Σ by polynomials with nonnegative coefficients. For X a real Banach space, K ⊆ X a closed but not necessarily normal cone with K − K = X, and A : X → X a bounded linear operator with A[K] ⊆ K, we use these approximation theorems to investigate when the spectral radius r(A) of A belongs to its spectrum σ(A). A special case of our results is that if X is a Hilbert space, A is normal and the 1-dimensional Lebesgue measure of σ(i(A − A *)) is zero, then r(A) ∈ σ(A). However, we also give an example of a normal operator A = −U − αI (where U is unitary and α > 0) for which A[K] ⊆ K and r(A) / ∈ σ(A).

Positive Semigroups of Kernel Operators

Positivity, 2008

Extending results of Davies and of Keicher on p we show that the peripheral point spectrum of the generator of a positive bounded C0-semigroup of kernel operators on L p is reduced to 0. It is shown that this implies convergence to an equilibrium if the semigroup is also irreducible and the fixed space non-trivial. The results are applied to elliptic operators.