A remark on cosine families

Abstract

Abstract.Let C(t), t 6 R, be a strongly continuous cosine family and A its def infinitesimal generator. Then the set E = {x e X: C{t)x is once continuously differentiable in t on R} of the Banach space X is contained in the domain of {-Af for 0 < a < 1/2.The purpose of this note is to prove for a strongly continuous cosine family C(t), t £ R, defined on a Banach space X and with infinitesimal generator A, that the def set E = {x £ X: C(t)x is once continuously differentiable in t on R} is containedis the domain of the a power of the operator -A.A one parameter family C(t), t £ R, of bounded linear operators mapping the Banach space X into itself is called a strongly continuous cosine family if and only if C(t + s) + C(t -s) = 2C(t)C(s) for all s,t £ R, C(0) = I, C(t)x is continuous in t £ R for each fixed x £ X. The associated sine family is given by S(t)x = f0 C(s)x ds for x £ X and t £ R. The linear operator A: X ^ X defined by Ax = C"(0)x and with dense domain D(A) = {x £ X: C(t)x is twice continuously differentiable in / on R} is called the infinitesimal generator of C(t), t £ R. For other properties of cosine families used in this paper see [2] or [9].The following theorem appears in C. Travis and G. Webb [8] and [9]: Theorem 1. Let C(t), t £ R, be a strongly continuous cosine family with associated sine family S(t), t £ R, and infinitesimal generator A. The following statements are equivalent:(i) there exists a closed linear operator B on X such that B2 = A and B commutes with every operator in B(X, X) which commutes with A ; S(t) maps X into D(B) for each t £ R; BS(t)x is continuous in t £ Rfor each fixed x £ X;(ii) E = D(B), the domain of B.The conditions stated in part (i) of Theorem 1 have also been considered by H. Fattorini in [2] and [3]. Fattorini has shown in [3] that every strongly continuous cosine family defined on the Banach space Lp, 1 <p < oo, satisfies condition (i).