2 EXAMPLES OF LOCAL ERGODIC DIVERGENCE

This note contains the first example of a 1-parameter semigroup {T pt≧0} of linear contractions in L p (1<p<∞) for which the assertion of the local ergodic theorem (t 1∝0′T sfds conv. a.e. as t → 0+0 for all f ∈L p) fails to be true. The first example is a continuous semigroup of unitary operators in L 2, the second a power-bounded continuous semigroup of positive operators in L 1. This answers problems of Kubokawa, Fong and Sucheston. © 1979 Hebrew University.