On the generalized semigroup relation in the strong operator topology

AbstractLet be a Banach space, and let –() denote the Banach algebra of endomorphisms of . A one-parameter family of operator-valued functions {S(t), t∈R+}, where S(t):R+→–(), is said to be a generalized semigroup of operators on if (1) S(s+t)−−S(s)S(t)=F(s, t), s, t∈R+, (2) S(s)S(t)=S(t)S(s), (3) S(O)=I, where F(s,t): R+× ×R+ → –(). Solutions of Eq. (1) in the vmiform operator topology were considered in an earlier paper of the authors (Proc. Nat. Acad. Sci. U.S.A. 60 (1968), 1170–1174). In this paper the authors investigate the analytical properties of Eq. (1) in the strong operator topology, when it is assumed that the perturbation family F(s, t) is bounded relative to the family S(t). A representation of S(t) is given; and, as an example, a generalized semigroup of translations on C[0, ∞] is considered