A QUANTITATIVE ASYMPTOTIC THEOREM FOR CONTRACTION-SEMIGROUPS WITH COUNTABLE UNITARY SPECTRUM

Let T be a semigroup of linear contractions on a Banach space X, and l et X(s)(T) = {z is an element of X : lim(s-->infinity) parallel to T(s )x parallel to = 0}. Then X(s)(T) is the annihilator of the bounded tr ajectories of T. If the unitary spectrum of T is countable, then X(s) (T) is the annihilator of the unitary eigenvectors of T, and lim(s) p arallel to T(s)x parallel to = inf{parallel to x - y parallel to : y i s an element of X(s)(T)} for each x in X.