Tohoku Mathematical Journal
2002
March
SECOND SERIES VOL. 54, NO. 1

Tohoku Math. J.
54 (2002), 43-59

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

Dedicated to Professor Satoru Igari on the occasion of his retirement from Tohoku University
Author Shigeru Hasegawa and Ryotaro Sato

(Received March 2, 2000, revised December 21, 2000)
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.

2000 Mathematics Subject Classification. Primary 47A35.
Key words and phrases. Pointwise local ergodic theorem, $(C_0)$-semigroup of linear contractions, Banach space of finite-dimensional-Hilbert-space-valued functions.

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