ON A PRODUCT FORMULA FOR UNITARY GROUPS

Abstract

For any nonnegative self-adjoint operators $A$ and $B$ in a separable Hilbert space, the Trotter-type formula $[;(e^{i2tA/n}+e^{i2tB/n})/2]^n$ is shown to converge strongly in the norm closure of $\dom(A^{1/2})\cap\dom(B^{1/2})$ for some subsequence and for almost every $t\in\mathbb{R}$. This result extends to the degenerate case, and to Kato-functions following the method of T. Kato (see ‘Trotter's product formula for an arbitrary pair of self-adjoint contraction semigroup’, Topics in functional analysis (ed. M. Kac, Academic Press, New York, 1978) 185–195). Moreover, the restrictions on the convergence can be removed by considering functions other than the exponential.