Order-bounded operators in vector lattices and in spaces of measurable functions

Abstract

The survey is devoted to the presentation of the state of the art of a series of directions of the theory of order-bounded operators in vector lattices and in spaces of measurable functions.The theory of disjoint operators, the generalized Hewitt-Yosida theorem, the connection with p-absolutely summing operators are considered in detail.

Calderón–Lozanovskii interpolation on quasi-Banach lattices

“…The class of (p, q)-regular operators was introduced in [2] (see also [3,13]), and has obvious connections with convexity and concavity (cf. [14, 1.d]).…”

(p, q)-Regular operators between Banach lattices

“…(1) For p = 2, the equalities given in (3) show that L 2 (µ) ⊗ α L 2 (ν) cannot be isomorphic to L 2 (µ) ⊗ r2,2 L 2 (ν) for α = d * 2 = d 2 = ∆ 2 = g * 2 = g 2 , since by Corollary 3.10, we have that r 2,2 is equivalent to π in this tensor product.…”

“…, for every choice of vectors {x i } n i=1 . The operators for which this inequality holds are called (p, q)-regular, and as far as we know were introduced by A. Bukhvalov in [2] in connection with the interpolation of Banach lattices (see also [3]). The aim of this note is to make a systematic study of the class of (p, q)-regular operators.…”