Sequences of contractions on L1-spaces

Abstract

Let T n , n = 1,2,… be a sequence of linear contractions on the space where is a finite measure space. Let M be the subspace of L 1 for which T n g → g weakly in L 1 for g ϵ M. If T n 1 → 1 strongly, then T n f → f strongly for all f in the closed vector sublattice in L 1 generated by M. This result can be applied to the determination of Korovkin sets and shadows in L 1. Given a set G ⊂ L 1, its shadow S( G) is the set of all f ϵ L 1 with the property that T n f → f strongly for any sequence of contractions T n , n = 1, 2,… which converges strongly to the identity on G; and G is said to be a Korovkin set if S( G) = L 1. For instance, if 1 ϵ G, then, where M is the linear hull of G and B M is the sub-σ-algebra of B generated by { x ϵ X: g( x) > 0} for g ϵ M. If the measure algebra is separable, has Korovkin sets consisting of two elements.