# Trotter product formula and linear evolution equations on Hilbert spaces On the occasion of the 100th birthday of Tosio Kato

## TL;DR

This paper investigates the approximation of solutions to linear evolution equations on Hilbert spaces using Trotter product formulas, establishing convergence conditions and rates based on operator regularity and Hölder continuity.

## Contribution

It provides new conditions under which the Trotter product formula converges for evolution equations with time-dependent operators on Hilbert spaces, including explicit convergence rates.

## Key findings

- Convergence of the approximation when Hölder exponent exceeds 2α - 1
- Operator norm approximation of solution operators by semigroup products
- Explicit convergence rate tied to Hölder continuity

## Abstract

The paper is devoted to evolution equations of the form $\partial$ $\partial$t u(t) = --(A + B(t))u(t), t $\in$ I = [0, T ], on separable Hilbert spaces where A is a non-negative self-adjoint operator and B($\times$) is family of non-negative self-adjoint operators such that dom(A $\alpha$) $\subseteq$ dom(B(t)) for some $\alpha$ $\in$ [0, 1) and the map A --$\alpha$ B($\times$)A --$\alpha$ is H{\"o}lder continuous with the H{\"o}lder exponent $\beta$ $\in$ (0, 1). It is shown that the solution operator U(t, s) of the evolution equation can be approximated in the operator norm by a combination of semigroups generated by A and B(t) provided the condition $\beta$ > 2$\alpha$ -- 1 is satisfied. The convergence rate for the approximation is given by the H{\"o}lder exponent $\beta$. The result is proved using the evolution semigroup approach.