The Fredholm index of a pair of commuting operators

Abstract

This paper concerns Fredholm theory in several variables, and its applications to Hilbert spaces of analytic functions. One feature is the introduction of ideas from commutative algebra to operator theory. Specifically, we introduce a method to calculate the Fredholm index of a pair of commuting operators. To achieve this, we define and study the Hilbert space analogs of Samuel multiplicities in commutative algebra. Then the theory is applied to the symmetric Fock space. In particular, our results imply a satisfactory answer to Arveson's program on developing a Fredholm theory for pure $d$-contractions when $d=2$, including both the Fredholmness problem and the calculation of indices. We also show that Arveson's curvature invariant is in fact always equal to the Samuel multiplicity for an arbitrary pure d-contraction with finite defect rank. It follows that the curvature is a similarity invariant.