Charles J. K. Batty

Tauberian theorems and stability of one-parameter semigroups

The main result is the following stability theorem: Let T = (T(t))tzo be a bounded Co-semigroup on a reflexive space X. Denote by A the generator of T and by a(A) the spectrum of A. If a(A) n iR is countable and no eigenvalue of A lies on the imaginary axis, then limt BOO T(t)x = O for all x E X.

ASYMPTOTICALLY ALMOST PERIODIC SOLUTIONS OF INHOMOGENEOUS CAUCHY PROBLEMS ON THE HALF-LINE

Let u be a bounded, uniformly continuous, mild solution of an inhomogeneous Cauchy problem on R + : u«(t) fl Au(t)›u(t )( t & 0). Suppose that u has uniformly convergent means, r(A) fiR is countable, and u is asymptotically almost periodic. Then u is asymptotically almost periodic. Related results have been obtained by Ruess and Vu4 , and by Basit, using dierent methods. A direct proof is given of a Tauberian theorem of Batty, van Neerven and Ra

Stability of individual elements under one-parameter semigroups

biger, and applications to Volterra equations are discussed.

Tauberian theorems and stability of solutions of the Cauchy problem

Let {T(t): t > O} be a C0-semigroup on a Banach space X with generator A, and let x E X. If a(A) n iR is empty and t 4 T(t)x is uniformly continuous, then IIT(t)xII -+ 0 as t -+ oo. If the semigroup is sun-reflexive, a(A) n iR is countable, Pa(A) n iR is empty, and t 4 T(t)x is uniformly weakly continuous, then T(t)x -0 weakly as t --oo . Questions of almost periodicity and of stabilization of contraction semigroups on Hilbert space are also discussed.

Local spectra and individual stability of uniformly bounded ₀-semigroups

Let f : R+ → X be a bounded, strongly measurable function with values in a Banach space X, and let iE be the singular set of the Laplace transform f in iR. Suppose that E is countable and α ∥∥∫∞ 0 e −(α+iη)uf(s + u) du ∥∥ → 0 uniformly for s ≥ 0, as α↘ 0, for each η in E. It is shown that ∥∥∥∥∫ t 0 e−iμuf(u) du− f(iμ) ∥∥∥∥→ 0 as t → ∞, for each μ in R \ E; in particular, ‖f(t)‖ → 0 if f is uniformly continuous. This result is similar to a Tauberian theorem of Arendt and Batty. It is obtained by applying a result of the authors concerning local stability of bounded semigroups to the translation semigroup on BUC(R+,X), and it implies several results concerning stability of solutions of Cauchy problems.

Domination and ergodicity for positive semigroups

We study the asymptotic behaviour of individual orbits T (·)x of a uniformly bounded C0-semigroup {T (t)}t≥0 with generator A in terms of the singularities of the local resolvent (λ− A)−1x on the imaginary axis. Among other things we prove individual versions of the Arendt-Batty-Lyubich-Vũ theorem and the Katznelson-Tzafriri theorem.

The Spectral Function and Principal Eigenvalues for Schrödinger Operators

In this note we study the conditions under which ergodicity is preserved by domination. As an application, the criterion for almost periodicity given by Ljubich-Phong [9] (see also Batty-Phong [4]) can be simplified for positive semigroups. 1. ERGODICITY AND DOMINATION Let T = (T(t))t>o be a positive Co-semigroup on a Banach lattice E with generator A. As usual, we say that T is mean-ergodic (or C-ergodic) if C limtoo T(t) := limto,0 t fof T(s) ds exists strongly. Due to the positivity of T, this is equivalent to the following two conditions (see [3]): (1.1) (0, ??) C p(A) and sup IIAR(A, A)II < oo o

Principal Eigenvalues and Perturbation

AbstractLet m ∈

Weighted and local stability of semigroups of operators

L_{loc}^1 (\mathbb{R}^N )