Fredholm operators associated with strongly pseudoconvex domains in Cn

Abstract

This paper generalizes the index theorem of Gohberg and Krien on Weiner-Hopf operators on the unit circle. Let Ω be a strongly pseudoconvex domain in C n and suppose L 2 N(Ω) is the space of square integrable functions ƒ: Ω → C N . Let H 2 N(Ω) be the subspace of all ƒ ϵ L 2 N(Ω) which are holomorphic in Ω and let P: L 2 N(Ω) → H 2 N(Ω) be the orthogonal projection. Let s be a continuous, N × N matrix valued function on Ω which is smooth in Ω such that det s(z) ≠ 0 for z ϵ ∂Ω , and let S : H 2 N(Ω) → L 2 N(Ω) be the operator defined by S 1ƒ = sƒ for ƒ ϵ H 2 N(Ω) . It is then proved that the operator S = PS 1 is Fredholm. The problem of obtaining a general formula for index is open. However, if Ω is the open unit ball in C n , then it is proved that the index of the operator S is equal to (−1) n time degree of the restriction of s to ∂Ω, if N ⩾ n. These results have obvious points of contact with the Atiyah-Singer index theorem.