A general Trotter–Kato formula for a class of evolution operators

In this article we prove new results concerning the existence and various properties of an evolution system UA+B(t, s)0 s t T generated by the sum −(A(t) + B(t)) of two linear, time-dependent and generally unbounded operators defined on time-dependent domains in a complex and separable Banach space B. In particular, writing L(B) for the algebra of all linear bounded operators on B, we can express UA+B(t, s)0 s t T as the strong limit in L(B) of a product of the holomorphic contraction semigroups generated by −A(t) and −B(t), respectively, thereby proving a product formula of the Trotter–Kato type under very general conditions which allow the domain D(A(t) + B(t)) to evolve with time provided there exists a fixed set D ⊂ t∈[0,T ]D(A(t) + B(t)) everywhere dense in B. We obtain a special case of our formula when B(t) = 0, which, in effect, allows us to reconstruct UA(t, s)0 s t T very simply in terms of the semigroup generated by −A(t). We then illustrate our results by considering various examples of nonautonomous parabolic initial–boundary value problems, including one related to the theory of timedependent singular perturbations of self-adjoint operators. We finally mention what we think remains an open problem for the corresponding equations of Schrödinger type in quantum mechanics. © 2009 Elsevier Inc. All rights reserved.