The Behavior of the Radical of the Algebras Generated by a Semigroup of Operators on Hilbert Space

Abstract

Let T = {T(t)}(t >= 0) be a continuous semigroup of contractions on a Hilbert space. We define A(T) as the closure of the set {(f) over cap (T) : f is an element of L-1 (R+)} with respect to the operator-norm topology, where (f) over cap (T) = [GRAPHICS] is the Laplace transform of f is an element of L-1 (R+) with respect to the semigroup T. Then, A(T) is a commutative Banach algebra. In this paper, we obtain some connections between the radical of A(T) and the set {R is an element of A(T) : T(t)R --> 0, strongly or in norm, as t --> infinity}. Similar problems for the algebras generated by a discrete semigroup {T-n : n = 0, 1, 2,...} is also discussed, where T is a contraction.