| Abstract: | 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 G of Green operators of order and type zero is a spectrally invariant Fréchet subalgebra of L(H), H a suitable Hilbert space, i. e.,
 - 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 G in several complex variables.
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