Lebesgue-type convergence theorems in Banach lattices with applications to compact operators

Abstract

The main result is a Lebesgue-type convergence theorem in the setting of Banach lattices, of which the classical Lebesgue dominated convergence theorem is the prime example. It is proved that whenever E is a Banach lattice which is an ideal in a Riesz space M, M having the principal projection property and the Egoroff property, and if a sequence in E is order convergent in M, then the sequence is norm convergent in E if it is contained in a set of uniformly absolutely continuous norm. This result enables one to derive compactness criteria for bounded linear operators on Banach lattices. If T: E → E ( E, F Banach lattices) is a norm bounded linear operator with T[E] ⊂ F a and if U denotes the unit ball of E, then T is compact if and only if (a) T[ U] is of uniformly absolutely continuous norm; (b) For every sequence S ⊂ T[U] the band generated by S in F a is an ideal in a Dede-kind complete Riesz space M with the Egoroff property and S has a subsequence converging in order in M to an element of M. From this theorem all the known compactness criteria for order bounded linear operators on Banach lattices can be derived. In particular the well-known compactness conditions proved by W. A. J. Luxemburg and A. C. Zaanen (Math. Annalen 149, 150–180 (1963)) for kernel operators on Banach function spaces are generalized to Banach lattices.