Compact integral operators on Banach function spaces

Abstract

In the monograph [8] of Halmos and Sunder, it is shown that if (X,/~), (Y, v) are finite measure spaces, then each integral operator T:U(Y,, v)~L2(X,#) is compact if considered as a map from L2(Y, v) to LI(X, #). In the present note we shall present two much more general results on compactness of integral operators, each one of them containing the above so-called (2.1)-compactness as a special case. To be specific, our first result is that each integral operator from the Banach function Lpl to the Banach function space Lp2 is compact provided a(Lp2)<min(s(Lp,), 2), where o(respectively s) denotes the upper (respectively lower) indices introduced for Banach function spaces by Grobler in [7]. Our second result is that if Lp, and Lp2 are as before and Lo~_Lp3, then each integral operator from Lp~ into Lp2 is compact as a mapping from Lp~ into Lo3 provided p] is order continuous and o-(Lp3 ) <s(Lo2 ). The approach of the present paper is entirely different to that of [8] and is partly based on a compactness criterion given in [4] which derives from a characterization due to Luxemburg and Zaanen [13] for compactness of order-bounded kernel operators on Banach function spaces. The methods of [4] not only yield the (2.1)-compactness result as a special case, but place it in perspective with reference to related work of And6 [1], who treated order-bounded kernel operators on Orlicz spaces, and Rosenthal [15], who considered continuous linear mappings from (subspaces of) LP-spaces. It is perhaps appropriate to point out that these latter papers have a common antecedent in the work of Pitt [14] for sequence spaces. We refer the reader to [19] for the basic facts and terminology of Banach function spaces and we shall use the monograph [16] as general reference on Riesz spaces and Banach lattices. We shall use L~ #) to denote the space of equivalence classes of realvalued #-measurable functions. If Lp is a Banach function space, then Ep denotes the first (K6the) associate space and (/2p) + shall denote the set of non-negative elements of Ep. The result contained in Theorem 2.7 was announced by the first author at the Oberwolfach meeting on