Let be a measure space and the usual Banach spaces. A linear operator T : Lp → Lpis called a positive contraction if it transforms non-negative functions into non-negative functions and if its norm is not more than one. The purpose of this note is to show that if 1 < p < ∞ and if T : Lp → Lp is a positive contraction then

Throughout this article p denotes a fixed number such that 1 ≤ p < ∞. The definition of a real Lp space associated with a measure space is well known. These spaces are Banach Spaces and, with the usual partial ordering of (equivalence classes of) functions, also Banach Lattices.

No abstract

1996). The strong sweeping out property for lacunary sequences, Riemann sums, convolution powers, and related matters.Abstract. In this paper we establish conditions on a sequence of operators which imply divergence. In fact, we give conditions which imply that we can find a set B of measure as close to zero as we like, but such that the operators applied to the characteristic function of this set have a lim sup equal to 1 and a lim inf equal to 0 a.e. (strong sweeping out). The results include the fact that ergodic averages along lacunary sequences, certain convolution powers, and the Riemann sums considered by Rudin are all strong sweeping out. One of the criteria for strong sweeping out involves a condition on the Fourier transform of the sequence of measures, which is often easily checked. The second criterion for strong sweeping out involves showing that a sequence of numbers satisfies a property similar to the conclusion of Kronecker's lemma on sequences linearly independent over the rationals.

Let (X, J, p) be a finite measure space. An invertible transformation