If T = {T (t); t ≥ 0} is a strongly continuous family of bounded linear operators between two Banach spaces X and Y and f ∈ L1(0, b, X), the convolution of T with f is defined by (T ∗f)(t) = ∫ t 0 T (s)f(t− s)ds. It is shown that T ∗ f is continuously differentiable for all f ∈ C(0, b, X) if and only if T is of bounded semi-variation on [0, b]. Further T ∗ f is continuously differentiable for a...

Fountain and Gomes have shown that any proper left ample semigroup embeds into a so-calledW -product, which is a subsemigroup of a reversed semidirect product T ⋉ Y of a semilattice Y by a monoid T , where the action of T on Y is injective with images of the action being order ideals of Y. Proper left ample semigroups are proper left restriction, the latter forming a much wider class. The aim o...

Pontryagin’s maximum principle in its infinite dimensional version provides (separate) necessary and su cient conditions for both time and norm optimality for the system y0 = Ay + u (A the infinitesimal generator of a strongly continuous semigroup); in particular it provides a costate z(t) for every time or norm optimal control ū(t) hitting a target ȳ 2 D(A). This paper shows that for the right...

Perturbations (r τt)t≥0 of the semigroup of shifts (τt)t≥0 on L(R+) are studied under the assumption that r τt−τt belongs to a certain Schatten–von Neumann class Sp with p ≥ 1. It is shown that, for the unitary component in the Wold– Kolmogorov decomposition of the cogenerator of the semigroup (r τt)t≥0, any singular spectral type may be achieved by S1-perturbations. An explicit construction is...

For convenience, we can have a look at the list of materials contained in this lecture first. Note here we always consider the time-homogenous Markov processes. • Transition kernels μt, t ≥ 0 ←→ transition operators semigroup Tt, t ≥ 0. • Feller semigroup Tt, t ≥ 0 • Feller processes • existence • every Feller process has a cadlag version • regularization theorem for submartingale • strong Mark...

Let $S$ be a semitopological semigroup. The $wap-$ compactification of semigroup S, is a compact semitopological semigroup with certain universal properties relative to the original semigroup, and the $Lmc-$ compactification of semigroup $S$ is a universal semigroup compactification of $S$, which are denoted by $S^{wap}$ and $S^{Lmc}$ respectively. In this paper, an internal construction of ...

We address the problems of extrapolation, analyticity, and Lp-spectral independence for C0-semigroups in the abstract context of metric spaces with exponentially bounded volume. The main application of the abstract result is Lp-analyticity of angle π 2 of Schrödinger semigroups on Riemannian manifolds with Ricci curvature bounded below, under the condition of form smallness of the negative part...

In this note we give an alternative construction of free groups by working with semigroups. We also give some basic properties of free groups. A semigroup is a nonempty set with an associative binary operation. If S is a semigroup, we denote the operation by (a, b) 7→ ab. If S has an identity, then we call S a semigroup with 1. Such semigroups are often called monoids. We will restrict our atte...