Fréchet 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 on (possibly) noncompact manifolds. It is based on a class of weighted symbols and Sobolev spaces. If the underlying manifold is compact, one recovers the standard calculus. The following is proven: (1) The algebra script G sign of Green operators of order and type zero is a spectrally invariant Fréchet subalgebra of ℒ(H), H a suitable Hubert space, i. e., script G sign ∩ ℒ(H)^-1 = script G sign^-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 modulo lower order symbols, and (4) There is a holomorphic functional calculus for the elements of script G sign in several complex variables.