ON A LOCAL ERGODIC THEOREM FOR FINITE-DIMENSIONAL-HILBERT-SPACE-VALUED FUNCTIONS

Abstract

We consider a Banach space of finite-dimensional-Hilbert-space-valued functions on a sigma-finite measure space. The norm of the function space is assumed to satisfy some suitable conditions. Then we prove a pointwise local ergodic theorem for a $(C_0)$-semigroup of linear contractions on the function space, under an additional norm condition for operators of the semigroup. Our result extends Baxter and Chacon's local ergodic theorem for scalar-valued functions.