Dunford-Pettis and strongly Dunford-Pettis operators on L1(μ)

Motivated by a problem in mathematical economics [4] Gretsky and Ostroy have shown [5] that every positive operator T:L1[0, 1] → c0 is a Dunford-Pettis operator (i.e. T maps weakly convergent sequences to norm convergent ones), and hence that the same is true for every regular operator from L1[0, 1] to c0. In a recent paper [6] we showed the converse also holds, thereby characterizing the D–P operators by this condition. In each case the proof depends (as do so many concerning D–P operators on Ll[0, 1]) on the following well-known result (see, e.g., [2]): If μ is a finite measure, an operator T:L1(μ) → E is a D–P operator is compact, where i:L∞(μ) → L1(μ) is the canonical injection of L∞(μ) into L1(μ). If μ is not a finite measure this characterization of D–P operators is no longer available, and hence results based on its use (e.g. [5], [6]) do not always have straightforward extensions to the case of operators on more general L1(μ) spaces.