The G-fredholm Property of the ∂̄-neumann Problem

Let H1 and H2 be Hilbert spaces and let B(H1,H2) be the space of bounded linear operators A : H1 → H2. An operator A ∈ B(H1,H2) is said to be Fredholm if first, the kernel ofA is finite-dimensional, and second the image ofA is closed and has finite codimension. An application of the open mapping theorem shows that the closedness requirement on the image is redundant. A well-known example of Fredholm operators (F. Riesz): if C is a compact operator then 1−C is Fredholm. It is easy to see that the Fredholm property is equivalent to invertiblility modulo finite-rank operators or compact operators. For a Fredholm operator A its index is defined by