On the Representation of Order Continuous Operators by Random Measures

Abstract

Abstract. Using the representation Tf(y) = ¡f'dvy, where (vy) is a random measure, we characterize some interesting bands in the lattice of all order-continuous operators on a space of measurable functions. For example, an operator T is (lattice-)orthogonal to all integral operators (i.e. all cv are singular) or belongs to the band generated by all Riesz homomorphisms (i.e. all r, are atomic) if and only if T satisfies certain properties which are modeled after the Riesz homomorphism property [31] and continuity with respect to convergence in measure. On the other hand, for operators orthogonal to all Riesz homomorphisms (i.e. all vy are diffuse) we give characterizations analogous to the characterizations of Dunford and Pettis, and Buhvalov for integral operators. The latter results are related to Enflo operators, to a result of J. Bourgain on Dunford-Pettis operators and martingale representations of operators.

Rademacher bounded families of operators on 𝐿₁

“…We will utilize the so-called random measure representation of operators on L 1 , developed in [6], [4] and [10].…”

Surjective Isometries on Rearrangement-Invariant Spaces

“…Remark: Note that (2) says that T : X → L 1 (|h|dµ) is an order-continuous operator for every h ∈ X ′ ; see Weis [29]. (2) → (1) : We must show that the adjoint T * : X * → X * maps X ′ into X ′ .…”

On Some Classes of Operators Determined by the Structure of Their Memory

“…It seems, however, that only the notion of disjointness-preserving (D-) operators was rather successful in this sense. Moreover, even with the latter notion, most significant results were obtained only for the linear operator case [20], while we are inclined to think (and try to justify in this paper) that no reasonably 'nice' properties can be proved for general D-operators in the nonlinear case.…”