Some connections between Pettis integration and operator theory
Elizabeth M. Bator, Paul Lewis, David Race · Rocky Mountain Journal of Mathematics · 1987
Introduction.Suppose that X is a Banach space with continuous dual X*, (fi, E,/i) is a finite measure space, / : U -• X is a scalarly measurable function so that x*f G L x (fi) for each x* € X*, and Tf : X* -> L 1 (/i) is the operator defined by Tf(x*) = x*/.In recent papers Huff [9] and Bator [2] effectively used properties of the operator Tf to study the Pettis integral.In this paper we extend this study to operators between general Banach spaces.In particular, we present a characterization of (w*,w)-continuous linear transformations, as well as new expositions of the Riddle, Saab, and Uhi result on universal Pettis integrability [16] and Odell's characterization (in terms of completely continuous operators) of spaces which contain i 1 [18].Throughout the paper, X and Y will denote real Banach spaces.We write X « Y to denote that X and Y are isomorphic (= linearly homeomorphic), and we denote the unit ball of X by Bx-By an operator T from X to y we shall mean a continuous linear transformation T : X -• Y; the adjoint of T will be denoted by T*.An operator T : X* -• Y is said to be (w*, w;)-continuous provided that (T(x*)) converges to T(x*) in the weak topology of Y whenever (x* ) is a net which converges to x* in the weak* topology of X*.We denote weak (weak*) convergence by -•(-•).If F is a finite subset of X and e > 0, set2. (w*,w)-Continuity.If (fi, E,//) is as above and / : Q -• X is a function, then we say that / is scalarly measurable with respect to ß if x*/ is /i-measurable for x* G X*, and we say that / belongs to weak L x (fi,X) if x*/ G L 1 ^) for all x* G X*.If / G weak-L^X), then we define the operator T f : X* -» L 1 ^) by T/(x*) = x*/ ([4], p. 52), and we say that / is a //-Pettis integrable if T* maps L°°{ii) into the canonical image of X in X**.In [9] Huff gave a simple proof