Chernoff and Trotter-Kato theorems for locally convex spaces

TL;DR

This paper generalizes Chernoff and Trotter-Kato theorems to locally convex spaces, extending previous results from Banach spaces and establishing new conditions for their validity.

Contribution

It introduces a new approach for the abstract Cauchy problem in locally convex spaces and proves generalized Chernoff, Lie-Trotter, and Trotter-Kato theorems with necessary and sufficient conditions.

Findings

Generalization of Chernoff and Trotter-Kato theorems to locally convex spaces

New necessary and sufficient conditions for product formulas

Extension of results to more general topological spaces

Abstract

We develop new approach for studying the abstract Cauchy problem $\dot{x}=Ax$, $x(0)=x_0\in D(A)$ for linear operators $A$ defined on a locally convex space $X$. This approach was firstly introduced in the paper "Chernoff and Trotter type product formulas" to study the problem for Banach spaces. In this paper we not only generalize the results of the previous paper to more general topological spaces but also get new results for Banach spaces. In particular, we prove the "local" extension of Chernoff-Trotter-Kato type theorems. Applying this result, we prove Chernoff, Lie-Trotter and Trotter-Kato theorems for locally convex spaces. Also we find necessary and sufficient conditions for the validity of the Chernoff and Trotter product formulas.