Vol. 22, No. 1, 1967

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The product formula for semigroups defined by Friedrichs extensions

William Guignard Faris

Vol. 22 (1967), No. 1, 47–70

Suppose that A and B are linear operators which generate semigroups on a Hilbert space. Then A + B may be far from being a generator. Nevertheless, a generator may sometimes be defined by adding operators corresponding to A and B, but with values in a larger Hilbert space, and then restricting the sum to the original Hilbert space. Here an explicit product formula in terms of the semigroups generated by A and B is shown to converge to a semigroup, which is that given by this sum. This result has application to the perturbation theory of partial differential equations. This is illustrated by the Feynman path integral representation of the solution of the Schrödinger equation with potential term containing singularities.

Mathematical Subject Classification
Primary: 47.50
Received: 28 June 1966
Published: 1 July 1967
William Guignard Faris