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

Xia Zhang, Leilei Wei, Ming Liu

TL;DR

The work extends Laplace transform theory to $\mathcal{T}_{\varepsilon,\lambda}$-complete random normed modules by leveraging both natural topologies, establishing differentiability, inversion, and uniqueness in the random setting. It then develops a full Hille–Yosida framework for exponentially bounded $C$-semigroups on RN-modules, covering both dense and nondense ranges of $C$, and shows how the $C$-resolvent arises from Laplace transforms of the semigroup. The results yield a rigorous link between semigroup dynamics and abstract Cauchy problems in random environments, including an explicit representation of solutions via the $C$-resolvent. This advances operator semigroup theory in random functional-analytic contexts and provides tools for well-posedness of abstract evolution equations under uncertainty.

Abstract

In this paper, we first introduce the notion of the Laplace transform for an abstract-valued function from $[0, \infty)$ to a $\mathcal{T}_{\varepsilon, λ}$-complete random normed module $S$. Then, combining respective advantages of the $(\varepsilon, λ)$-topology and the locally $L^0$-convex topology on $S$, we prove the differentiability, Post-Widder inversion formula and uniqueness of such a Laplace transform. Second, based on the above work, we establish the Hille-Yosida theorem for an exponentially bounded $C$-semigroup on $S$, considering both the dense and nondense cases of the range of $C$, respectively, which extends and improves several important results. Finally, we also apply such a Laplace transform to abstract Cauchy problems in the random setting.

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

TL;DR

The work extends Laplace transform theory to -complete random normed modules by leveraging both natural topologies, establishing differentiability, inversion, and uniqueness in the random setting. It then develops a full Hille–Yosida framework for exponentially bounded -semigroups on RN-modules, covering both dense and nondense ranges of , and shows how the -resolvent arises from Laplace transforms of the semigroup. The results yield a rigorous link between semigroup dynamics and abstract Cauchy problems in random environments, including an explicit representation of solutions via the -resolvent. This advances operator semigroup theory in random functional-analytic contexts and provides tools for well-posedness of abstract evolution equations under uncertainty.

Abstract

In this paper, we first introduce the notion of the Laplace transform for an abstract-valued function from to a -complete random normed module . Then, combining respective advantages of the -topology and the locally -convex topology on , we prove the differentiability, Post-Widder inversion formula and uniqueness of such a Laplace transform. Second, based on the above work, we establish the Hille-Yosida theorem for an exponentially bounded -semigroup on , considering both the dense and nondense cases of the range of , respectively, which extends and improves several important results. Finally, we also apply such a Laplace transform to abstract Cauchy problems in the random setting.

Paper Structure

This paper contains 7 sections, 18 theorems, 160 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Laplace transform for an abstract-valued function from $[0, \infty)$ to a $\mathcal{T}_{\varepsilon, \lambda}$-complete $R N$ module
  4. $C$-semigroups of continuous module homomorphisms on a $\mathcal{T}_{\varepsilon, \lambda}$-complete $R N$ module
  5. The Hille-Yosida theorem on a $\mathcal{T}_{\varepsilon, \lambda}$-complete $R N$ module $S$ for the case that $R(C)$ is nondense in $(S, \mathcal{T}_{\varepsilon, \lambda})$
  6. The Hille-Yosida theorem on a $\mathcal{T}_{\varepsilon, \lambda}$-complete $R N$ module $S$ for the case that $R(C)$ is dense in $(S, \mathcal{T}_{\varepsilon, \lambda})$
  7. An application to abstract Cauchy problems on a $\mathcal{T}_{\varepsilon, \lambda}$-complete $RN$ module

Key Result

proposition 1

Suppose that $S$ is a $\mathcal{T}_{\varepsilon, \lambda}$-complete $R N$ module and a function $g:[s, t] \rightarrow (S, \mathcal{T}_{\varepsilon, \lambda})$ is continuously differentiable. If $g$ is $L^0$-Lipschitz on $[s, t]$, then $g^{\prime}$ is Riemann integrable on $[s, t]$ and $\int_s^t g^{\

Theorems & Definitions (37)

  • definition 1: Guo01
  • definition 2: Guo01
  • definition 3: Guo10
  • proposition 1: Guozhang12
  • proposition 2: Guozhang12
  • proposition 3: Zhangliu13
  • definition 4
  • definition 5
  • Lemma 3.1
  • proof
  • ...and 27 more