The Fredholm index of a pair of commuting operators

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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.

Original languageEnglish
Pages (from-to)367-402
Number of pages36
JournalGeometric and Functional Analysis
Issue number2
StatePublished - Apr 2006


  • Commuting operators
  • Fredholm index
  • Hilbert polynomials
  • Samuel multiplicity
  • Symmetric Fock space


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