A SPLITTING THEOREM FOR THE SPACE OF SMOOTH FUNCTIONS

We will show using purely linear functional analytic methods that each exact complex 0 --> F --> (C-infinity(Omega))(s0) -->(P0) (C-infinity (Omega))(s1) -->(P1) (C-infinity(Omega))(s2) -->(P2)..., where p(i) ar e matrices of convolution operators (in particular, linear differentia l operators with constant coefficients), splits from p(1) on (in the c ategory of topological vector spaces). Moreover, we characterize when these complexes split completely. The obtained result covers all known particular cases as well as some more general cases independently of the analytic nature of p(i). The result is a consequence of studying s plitting of short exact sequences () 0 --> F -->(j) X -->(q) G --> 0 () where F, X G are graded Frechet spaces, i.e., Frechet spaces equip ped with fixed reduced spectra representing them and j, q are consiste nt with the graded structures. We characterize splitting of () in cas e: (i) F or G are spaces C-infinity(Omega), where Omega is an open sub set in R '', or (ii) G is the space of all sequences. (C) 1998 Academi c Press.