On a Product Formula for Unitary Groups

For any nonnegative self-adjoint operators A and B in a separableHilbert space, the Trotter-type formula is shown to converge strongly in the norm closureof dom (A^1/2) dom (B^1/2 for some subsequence and for almostevery t R. This result extends to the degenerate case, andto Kato-functions following the method of T. Kato (see ‘Trotter'sproduct formula for an arbitrary pair of self-adjoint contractionsemigroup’, Topics in functional analysis (ed. M. Kac,Academic Press, New York, 1978) 185–195). Moreover, therestrictions on the convergence can be removed by consideringfunctions other than the exponential. 2000 Mathematics SubjectClassification 47D03 (primary), 47B25 (secondary).