Samuel multiplicity and the structure of semi-Fredholm operators

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Abstract

Two numerical invariants refining the Fredholm index are introduced for any semi-Fredholm operator in such a way that their difference calculates the Fredholm index. These two invariants are inspired by Samuel multiplicity in commutative algebra, and can be regarded as the stabilized dimension of the kernel and cokernel. A geometric interpretation of these invariants leads naturally to a 4×4 uptriangular matrix model for any semi-Fredholm operator on a separable Hilbert space. This model can be regarded as a refined, local version of the Apostol's 3×3 triangular representation for arbitrary operators. Some classical results, such as Gohberg's punctured neighborhood theorem, can be read off directly from our matrix model. Banach space operators are also considered.

Original languageEnglish
Pages (from-to)411-437
Number of pages27
JournalAdvances in Mathematics
Volume186
Issue number2
DOIs
StatePublished - 20 Aug 2004

Keywords

  • Fredholm index
  • Hilbert module
  • Operator matrix
  • Samuel multiplicity
  • Semi-Fredholm operator

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