The Hille-Yosida generation theorem for almost surely bounded $C_{0}$--semigroups of continuous module homomorphisms$^1$

Intermediate and extrapolated spaces for bi-continuous semigroups

We discuss the construction of the full Sobolev (Hölder) scale for non-densely defined operators on a Banach space with rays of minimal growth. In particular, we give a construction for extrapolation- and Favard spaces of generators of (bi-continuous) semigroups, or which is essentially the same, Hille-Yosida operators on Saks spaces.

Generation of relatively uniformly continuous Semigroups on Vector Lattices

In this paper we prove a Hille-Yosida type theorem for relatively uniformly continuous positive semigroups on vector lattices. We introduce the notions of relatively uniformly continuous, differentiable, and integrable functions on $\mathbb{R}_+$. These notions allow us to study the generators of relatively uniformly continuous semigroups. Our main result provides sufficient and necessary conditions for an operator to be the generator of an exponentially order bounded, relatively uniformly continuous, positive semigroup.

Regularity properties of some perturbations of non-densely defined operators with applications

This paper is to study some conditions on semigroups, generated by some class of non-densely defined operators in the closure of its domain, in order that certain bounded perturbations preserve some regularity properties of the semigroup such as norm continuity, compactness, differentiability and analyticity. Furthermore, we study the critical and essential growth bound of the semigroup under bounded perturbations. The main results generalize the corresponding results in the case of Hille-Yosida operators. As an illustration, we apply the main results to study the asymptotic behaviors of a cla…

$C_0$-sequentially equicontinuous semigroups: theory and applications

We present and apply a theory of one parameter $C_0$-semigroups of linear operators in locally convex spaces. Replacing the notion of equicontinuity considered by the literature with the weaker notion of sequential equicontinuity, we prove the basic results of the classical theory of $C_0$-equicontinuous semigroups: we show that the semigroup is uniquely identified by its generator and we provide a generation theorem in the spirit of the celebrated Hille-Yosida theorem. Then, we particularize the theory in some functional spaces and identify two locally convex topologies that allow to gather u…

用拉普拉斯变换方法研究$C$-半群在$\mathcal{T}_{\varepsilon, λ}$-完备随机范数模上的性质

A Laplace transform approach to $C$-semigroups on a $\mathcal{T}_{\varepsilon, λ}$-complete random normed module

本文通过引入抽象函数的拉普拉斯变换，结合$(\varepsilon, λ)$拓扑与局部$L^0$-凸拓扑，证明了拉普拉斯变换的可微性、后-威德公式及唯一性，并扩展了Hille-Yosida定理至指数有界$C$-半群，同时应用于随机背景下的抽象柯西问题。

Strongly continuous and locally equi-continuous semigroups on locally convex spaces

We consider locally equi-continuous strongly continuous semigroups on locally convex spaces (X,tau). First, we show that if (X,tau) has the property that weak* compact sets of the dual are equi-continuous, then strong continuity of the semigroup is equivalent to weak continuity and local equi-continuity. Second, we consider locally convex spaces (X,tau) that are also equipped with a `suitable' auxiliary norm. We introduce the set N of tau continuous semi-norms that are bounded by the norm. If (X,tau) has the property that N is closed under countable convex combinations, then a number of Banach…