The Fredholm index of a pair of commuting operators, II

Xiang Fang

Research output: Contribution to journalArticlepeer-review

9 Scopus citations


We first show that an inequality on Hilbert modules, obtained by Douglas and Yan in 1993, is always an equality. This allows us to establish the semi-continuity of the generalized Samuel multiplicities for a pair of commuting operators. Then we discuss the general structure of a Fredholm pair, aiming at developing a model theory. For application we prove that the Samuel additivity formula on Hilbert spaces of holomorphic functions is equivalent to a generalized Gleason problem. As a consequence it follows the additivity of Samuel multiplicity, in its full generality, on the symmetric Fock space. During the course we discover that a variant e (ṡ) of the classic algebraic Samuel multiplicity might be more suitable for Hilbert modules and can lead to better results.

Original languageEnglish
Pages (from-to)1669-1692
Number of pages24
JournalJournal of Functional Analysis
Issue number6
StatePublished - 15 Mar 2009


  • A tuple of commuting operators
  • Fredholm index
  • Multivariable operator theory
  • Samuel multiplicity
  • Symmetric Fock space


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