Chernoff and Trotter type product formulas

TL;DR

This paper investigates the abstract Cauchy problem for linear operators on Banach spaces, proving solution uniqueness, representation via product formulas, and criteria for operator sums to generate C_0-semigroups.

Contribution

It establishes conditions for solution uniqueness, representation as limits of product formulas, and criteria for sums of generators to form new generators of C_0-semigroups.

Findings

Proved uniqueness of local solutions for a class of operators.

Represented solutions as limits of product formulas in the weak operator topology.

Provided criteria for the sum of two generators to be a generator of a C_0-semigroup.

Abstract

We consider the abstract Cauchy problem x'=Ax, x(0)=x_0\in D(A) for linear operators A on a Banach space X. We prove uniqueness of the (local) solution of this problem for a natural class of operators A. Moreover, we establish that the solution x(\cdot) can be represented as a limit of sequence F(t/n)^{n} as n\to\infty in the weak operator topology, where a function F:[0,\infty)\to L(X) satisfies F'(0)y=Ay, y\in D(A). As a consequence, we deduce necessary and sufficient conditions that a linear operator C is closable and its closure is a generator of C_0-semigroup. We also obtain some criteria for the sum of two generators of C_0-semigroups to be a generator of C_0-semigroup such that the Trotter formula is valid.