An Operator Bound Related to Regular Operators

Regular operators on L p-spaces are characterised in terms of an operator bound which is associated with certain generalisations of the Feynman-Kac formula. of multiplication by the characteristic functions of elements of the-algebra S, so that if f is a bounded S-measurable function, Q(f) = R f dQ is the operator of multiplication by f. When is it true that we can nd a positive number C such that the inequality (1) k X j=1 Q(g j)TQ(f j) C k X j=1 f j g j 1 ; holds for all bounded C-valued S-measurable functions f j ; g j ; j = 1; : : :; k; deened on and all k = 1; 2; : : :? Here k k is the operator norm on the space of bounded linear operators acting on L 2 ((; S; ; C), u v denotes the function (u v)(x; y) = u(x)v(y); for all (x; y) 2 R 2 , and the norm k k 1 denotes the L 1-norm with respect to on : Another way of stating inequality (1) is that the bilinear map (f; g) 7 ! Q(g)TQ(f); f; g 2 L 1 ((; S; ; C) is continuous for the topology of bi-equicontinuous convergence. The operator bound (1) arises in the construction of an operator valued set function M t from a semigroup S of operators and the spectral measure Q. If bounds of the form (1) hold, then the additive set function M t is bounded on an underlying algebra of sets, as outlined in 3]. It turns out that ideas arising in the theory of complex vector lattices are relevant to the characterisation of operators T for which the bound (1) holds, namely, regular operators. In J], an additional assumption was made that the operator T is a Fourier multiplier operator acting on an L p-space of a locally compact abelian group, in which case more properties than in the present situation can be established. However, operators which arise in applications are often not Fourier multi-plier operators. Section one introduces some terminology from vector lattices in the context of L p-spaces. The main result of the present note, Theorem 1, is presented in section