Frank Räbiger

Exponential stability, exponential expansiveness, and exponential dichotomy of evolution equations on the half-line

AbstractLetU=(U(t, s))t≥s≥O be an evolution family on the half-line of bounded linear operators on a Banach spaceX. We introduce operatorsGO,GX andIX on certain spaces ofX-valued continuous functions connected with the integral equation

Tauberian theorems and stability of solutions of the Cauchy problem

u(t) = U(t,s)u(s) + \int_s^t {U(t,\xi )f(\xi )d\xi }

Asymptotic properties of mild solutions of nonautonomous evolution equations with applications to retarded differential equations

, and we characterize exponential stability, exponential expansiveness and exponential dichotomy ofU by properties ofGO,GX andIX, respectively. This extends related results known for finite dimensional spaces and for evolution families on the whole line, respectively.

On the approximation of positive operators and the behaviour of the spectra of the approximants

(see e.g. [Da-K], [Fat], [Paz], [Tan]). In the following a family (U(t, s))(t,s)∈D in L(X) satisfying (E1)–(E3) is called an evolution family. It has been noticed by several authors (see [LM1], [LM2], [LRa], [Na2], [RaS], [Ra1], [Ra2], [Ra3], [Rha] and the references therein) that asymptotic properties of the evolution family (U(t, s))(t,s)∈D are strongly related to the asymptotic behaviour of an associated evolution semigroup (TE(t))t≥0 of operators on a Banach space E(X) of X –valued functions (see Section 1). For a large class of these function spaces this evolution semigroup is strongly continuous and hence has a generator GE . It has been shown by R. Rau [Ra1, Prop. 1.7] and Y. Latushkin and S. Montgomery–Smith [LM1, Thm. 3.1], [LM2, Thm. 4] that on the function spaces C0(IR, X) and L (IR, X), 1 ≤ p <∞ , these semigroups always satisfy the spectral mapping theorem

Superstable operators on Banach spaces

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.

Spectral and asymptotic properties of dominated operators

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.

Beiträge zur Strukturtheorie der Grothendieck-Räume

We investigate the asymptotic properties of the inhomogeneous nonautonomous evolution equation (d/dt)u(t)=Au(t)

Spectral Mapping Theorems For Evolution Semigroups on Spaces of Almost Periodic Functions

LetT be a positive linear operator on the Banach latticeE and let (Sn) be a sequence of bounded linear operators onE which converge strongly toT. Our main results are concerned with the question under which additional assumptions onSn andT the peripheral spectra πσ(Sn) ofSn converge to the peripheral spectrum πσ(T) ofT. We are able to treat even the more general case of discretely convergent sequences of operators.