Isometries of L ^p(X) and Vector Valued Ergodic Theorems

Abstract

Let X be a Banach space. We prove that surjective isometrics of L ^p(X), 1 < p < ∞, satisfy a vector valued maximal inequality and therefore (when X is reflexive) the vector valued ergodic theorem (the Cesaro means $(\frac{1}{n}~\sum\nolimits_{i=1}^{n}{{{T}^{i}}}~f){{~}_{n\in N}}$ converge a.s. in norm).

When X is a Banach lattice and T a positive isometry from L ^p(X) into itself, we give a representation of T, under appropriate convexity and concavity properties on X. This representation implies the same vector valued maximal inequality.

When X = L ^q, we give such a representation for all isometrics of L ^p(X).

We also prove that the dilation theorem of Akcoglu does not extend in a natural way to positive contractions of L ^p(L ^q).