Trotter Product Formula and Linear Evolution Equations on Hilbert Spaces

Abstract

The paper is devoted to evolution equations of the form $$\begin{aligned} \frac{\partial }{\partial t}u(t) = -(A + B(t))u(t), \quad t \in {\mathcal {I}}= [0,T], \end{aligned}$$ on separable Hilbert spaces where A is a non-negative self-adjoint operator and \(B(\cdot )\) is family of non-negative self-adjoint operators such that \(\mathrm {dom}(A^{\alpha }) \subseteq \mathrm {dom}(B(t))\) for some \({\alpha }\in [0,1)\) and the map \(A^{-{\alpha }}B(\cdot )A^{-{\alpha }}\) is Holder continuous with the Holder 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 Holder exponent \({\beta }\). The result is proved using the evolution semigroup approach.