A generalization of Chernoff's product formula for time-dependent operators

In this article we provide a set of sufficient conditions that allow a natural extension of Chernoff's product formula to the case of certain one-parameter family of functions taking values in the algebra L(B) of all bounded linear operators defined on a complex Banach space B. Those functions need not be contraction-valued and are intimately related to certain evolution operators U(t,s)0⩽s⩽t⩽T on B. The most direct consequences of our main result are new formulae of the Trotter–Kato type which involve either semigroups with time-dependent generators, or the resolvent operators associated with these generators. In the general case we can apply such formulae to evolution problems of parabolic type, as well as to Schrödinger evolution equations albeit in some very special cases. The formulae we prove may also be relevant to the numerical analysis of non-autonomous ordinary and partial differential equations.