On weakly compact operators on Banach lattices

Abstract

Consider a Banach lattice E and an order bounded weakly compact operator T: E -E. The purpose of this note is to study the weak compactness of operators that are related with T in some order sense. The main results are the following. (1) If T is a positive weakly compact operator and an operator S: E -+ E satisfies O < S 6 T, then S2 is weakly compact. (Examples show that S need not be weakly compact.) (2) If T and S are as in (1) and either S is an orthomorphism or E has an order continuous norm, then S is weakly compact. (3) If E is an abstract L-space and T is weakly compact, then the modulus I TI is weakly compact. For notation and terminology concerning Banach lattices we follow [1] and [5]. Consider a Banach lattice E and a positive weakly compact operator T: E -E. Now, if S: E -* E is an operator such that 0 < S < T holds, then what effect does the weak compactness of T have on S? Before giving some positive answers to this question, we shall present two examples to show that (in general) under these conditions S need not be a weakly compact operator. EXAMPLE 1. Let { r,J denote the sequence of Rademacher functions on [0, 1]. That is, rn(t) = Sgn sin(27st). Consider S, T: L,[O, 1] -* 1I defined by S(f) = (fof(x)r (x) dx) and T(f) = (fAf(x) dx, f f(x) dx, . . ). Then T is compact (it has rank one), and 0 < S < T holds. On the other hand, S is not a weakly compact operator. To see this, start by observing that the sequence