Let E be a real Banach space. We study the Ornstein-Uhlenbeck semigroup P = {P (t)}t≥0 associated with the Ornstein-Uhlenbeck operator Lf(x) = 1 2 TrQDf(x) + 〈Ax,Df(x)〉, x ∈ E. Here Q ∈ L (E, E) is a positive symmetric operator and A is the generator of a C0-semigroup S = {S(t)}t≥0 on E. Under the assumption that P admits an invariant measure μ∞ we prove that if S is eventually compact and the ...

Suppose A, S, and T are semigroups, e: A → S and f : A → T semigroup homomorphisms, and X a generating set for S. We assume (1) that every element of S divides some element of e(A), (2) that T is cancellative, (3) that T is power-cancellative (i.e, xd = yd ⇒ x = y for d > 0), and (4) a further technical condition, which in particular holds if T admits a semigroup ordering with the order-type of...

It is the fundamental principle of semigroup theory that the behavior of a strongly continuous semigroup (T (t))t≥0 on a Banach space X and the properties of its generator (A,D(A)), or equivalently the properties of the resolvent function R(λ,A) = (λ − A)−1 (λ ∈ C), should closely correlate. Indeed, the Laplace transform carries the regularity properties of the semigroup to the resolvent functi...

Let Ψ be a non-constant complex-valued analytic function defined on connected, open set containing the L p -spectrum of Laplacian ℒ homogeneous tree. In this paper we give necessary and sufficient condition for semigroup T(t)=e tΨ(ℒ) to chaotic -spaces. We also study dynamics t(aℒ+b) separately obtain sharp range b which T(t) is It includes some important semigroups such as heat Schrödinger sem...

There exist (C0) semigroups Tx(i), T2(t) on Hubert space with the following properties: 7f has a bounded generator and is uniformly bounded, but is not similar to a contraction semigroup. T2 is uniformly bounded, and there exists no scalar a such that e~"'T2(t) is similar to a contraction semigroup.

Let {Tt}t≥0 be a hypercyclic strongly continuous semigroup of operators. Then each Tt (t > 0) is hypercyclic as a single operator, and it shares the set of hypercyclic vectors with the semigroup. This answers in the affirmative a natural question concerning hypercyclic C0-semigroups. The analogous result for frequent hypercyclicity is also obtained.

Given a free additive convolution semigroup (?t)t?0 and probability measure ? on R, we find the necessary sufficient conditions for process ?t?? to be Lebesgue absolutely continuous with positive analytic density throughout R at all time t>0. For semigroups without this property, of its zeros. These results are quantified by Lévy semigroup, making it fairly easy construct many concrete examples...

In this paper we give some characterizations of topological extreme amenability. Also we answer a question raised by Ling [5]. In particular we prove that if T is a Borel subset of a locally compact semigroup S such that M(S)* has a multiplicative topological left invariant mean then T is topological left lumpy if and only if there is a multiplicative topological left invariant mean M on M(S)* ...

This paper is concerned with the initial value problem for a strictly hyperbolic n n system of conservation laws in one space dimension: u t + F(u) x = 0; u(0; x) = u(x): () Each characteristic eld is assumed to be either linearly degenerate or genuinely nonlinear. We prove that there exists a domain D L 1 , containing all functions with suuciently small total variation, and a uniformly Lipschi...