The Fredholm index of a pair of commuting operators |
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Authors: | Xiang Fang |
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Institution: | (1) Department of Mathematics, University of Alabama, Tuscaloosa, AL 35487, USA;(2) Present address: Department of Mathematics, Kansas State University, Manhattan, KS 66502, USA |
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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.
Received: October 2004 Revision: May 2005 Accepted: May 2005
Partially supported by National Science Foundation Grant DMS 0400509. |
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Keywords: | Fredholm index commuting operators Samuel multiplicity Hilbert polynomials symmetric Fock space |
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