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

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.