Frechet algebra techniques for boundary value problems on noncompact manifolds: Fredholm criteria and functional calculus via spectral invariance

A Boutet de Monvel type calculus is developed for boundary value problems o n (possibly) noncompact manifolds. It is based on a class of weighted symbo ls and Sobolev spaces. If the underlying manifold is compact, one recovers the standard calculus. The following is proven: (1) The algebra G of Green operators of order and type zero is a spectrally invariant Frechet subalgebra of L(H), H a suitable Hilbert space, i.e., G boolean AND L(H)(-1) = G(-1). (2) Focusing on the elements of order and type zero is no restriction since there are order reducing operators within the calculus. (3) There is a necessary and sufficient criterion for the Fredholm property of boundary value problems, based on the invertibility of symbols module l ower order symbols, and (4) There is a holomorphic functional calculus for the elements of G in sev eral complex variables.