A pr 2 00 4 Note on a product formula for unitary groups

For any nonnegative self-adjoint operators A and B in a separable Hilbert space, we show that the Trotter-type formula [(ei2tA/n + ei2tB/n)/2]n converges strongly in dom(A1/2) ∩ dom(B1/2) for some subsequence and for almost every t ∈ R. This result extends to the degenerate case and to Katofunctions following the method of Kato [6]. In a famous paper [6], T. Kato proved that for any nonnegative self-adjoint operators A and B in a Hilbert spaceH, the Trotter product formula (ee) converges strongly to the (degenerate) semigroup generated by the form-sum A+̇B for any t with Re t > 0. He also extended the result to a class of so-called Katofunctions, and to degenerate semigroups. However the convergence on the boundary iR remains an unclear problem in this generality [1, 3]. For Kato-functions f such that Im f ≤ 0 (for example f(s) = (1+ is)), Lapidus found such an extension [7]. For the case of the Trotter product formula with projector (eP ), Exner and Ichinose obtained recently a interesting result [4]. 1 Statement of the result Since this note is closely related to Kato’s paper [6], it is convenient to use similar notations. A and B denote nonnegative self-adjoint operators defined in closed subspaces MA and MB of a separable Hilbert space H, and PA, PB denote the orthogonal projections on MA and MB. Let D ′ = dom(A) ∩ dom(B), let H be the closure of D, and let P ′ be the orthogonal projector on H. The formsum C = A+̇B is defined as the self-adjoint operator in H associated with the nonnegative, closed quadratic form u 7→ ‖Au‖ + ‖Bu‖, u ∈ D. We consider Trotter-type product formulae F (t/n) based on the arithmetic mean F (t) = f(2tA)PA + g(2tB)PB 2 . (1) The Kato-functions f and g are assumed here to be bounded, holomorphic functions in {t ∈ C : Re t > 0} with: |f(t)| ≤ 1, f(0) = 1, f (+0) = lim t→0,Re t>0 f(t)− 1 t = −1, (2) and 0 ≤ f(s) ≤ 1 if s > 0, and the same conditions for g. By the functional calculus for normal operators, F (t) is well defined for Re t ≥ 0 and bounded by 1. Department of Theoretical Physics, quai Ernest-Ansermet 24, CH-1211 GENEVA 4 SWITZERLAND. email: [email protected]