C-Cosine Functions and the Abstract Cauchy Problem, II*

For a bounded linear injection C on a Banach space X and a closed linear operator A : D A.;XªX which commutes with C we prove that 1. the abstract Cauchy problem, u0 t.sAu t., tgR, u 0.sCx, u9 0.sCy, has a unique strong solution for every x, ygD A. if and only if 2. A1sA N D A2 . generates a C1-cosine function on X1 D A. with the graph norm., if and only if, in case A has nonempty resolvent set. 3. A generates a C-cosine function on X. Here C1sC N X1. Under the assumption that A is densely defined and Cy1ACs A, statement 3. is also equivalent to each of the following statements: 4. the problem ¨ 0 t.sA¨ t.qC xqty.qH0t Cg r . dr, tgR, ¨ 0.s¨ 9 0.s0, has a unique strong solution for every ggL1 and x, ygX; 5. the problem w0 t.s loc Aw t.qCg t., tgR, w 0.sCx, w9 0.sCy, has a unique weak solution for every ggL1loc and x, ygX. Finally, as an application, it is shown that for any bounded operator B which commutes with C and has range contained in the range of C, AqB is also a generator.