An Operator Bound Related to Feynman-kac Formulae

Those Fourier matrix multiplier operators which are convolutions with respect to a matrix valued measure are characterised in terms of an operator bound. As an application, the finite-dimensional distributions of the process associated with Dirac equation are shown to be unbounded on the algebra of cylinder sets. In its traditional incarnation, the Feynman-Kac formula is a means of expressing a perturbation to the heat semigroup in terms of an integral with respect to Wiener measure. It has been useful in proving estimates in quantum physics [Si], and an analogue of the formula is an important tool of quantum field theory [G-J]. Perturbations of the groups of operators associated with certain classes of hyperbolic differential equations can also be represented in terms of integrals with respect to tr-additive operator valued measures along the lines of the Feynman-Kac formula [12], [Jl]. To establish the existence of tr-additive operator valued measures associated with particular evolution equations, the following question arises. Suppose that E is a locally compact abelian group with a given Haar measure X. Let n = 1, 2,... and let T : L2(Z, C) -+ L2(Z, C") be a Fourier matrix multiplier operator acting on the space L2(2Z, C") of all ( A-equivalence classes of) functions square integrable with respect to X and with values in C". This means that if the Fourier transform of a function g e L2(L,Cn) is denoted by g, then there exists a bounded Borel measurable function Or : Y —> 5Ai 0, the inequality k (i) Y.Q(Sj)TQ(fj) j=i holds for all bounded scalar valued Borel measurable functions f, g¡, j Received by the editors April 2, 1993. 1991 Mathematics Subject Classification. Primary 47A30, 43A25; Secondary 81S40, 35L45. The author appreciates the hospitality of Macquarie University where the present work was completed. ©1994 American Mathematical Society 0002-9939/94 $1.00+ $.25 per page <c Y,fj®Sj ;=i