C-Cosine Functions and the Abstract Cauchy Problem, II

AbstractFor a bounded linear injectionCon a Banach spaceXand a closed linear operatorA:D(A)⊂X→Xwhich commutes withCwe prove that (1) the abstract Cauchy problem,u″(t)=Au(t),t∈R,u(0)=Cx,u′(0)=Cy, has a unique strong solution for everyx,y∈D(A) if and only if (2)A1=A∣D(A2) generates aC1-cosine function onX1(D(A) with the graph norm), if (and only if, in caseAhas nonempty resolvent set) (3)Agenerates aC-cosine function onX. HereC1=C∣X1. Under the assumption thatAis densely defined andC−1AC=A, statement (3) is also equivalent to each of the following statements: (4) the problemv″(t)=Av(t)+C(x+ty)+∫t0Cg(r)dr,t∈R,v(0)=v′(0)=0, has a unique strong solution for everyg∈L1locandx,y∈X; (5) the problemw″(t)=Aw(t)+Cg(t),t∈R,w(0)=Cx,w′(0)=Cy, has a unique weak solution for everyg∈L1locandx,y∈X. Finally, as an application, it is shown that for any bounded operatorBwhich commutes withCand has range contained in the range ofC,A+Bis also a generator