The Fredholm index of quotient Hilbert modules
type: Item
instance of: scholarly article
title: The Fredholm index of quotient Hilbert modules
zbMATH Open document ID: 1100.47009
author: Xiang Fang
published in: Mathematical Research Letters
publication date: 2006-01-24T00:00:00Z
review text: Let \(H\) be a Hilbert module in the sense of a Hilbert space equipped with a module structure over the polynomial ring \(A = C[z_1, \dots , z_n]\) in \(n\) complex variables such that the action by each coordinate function \(z_i\) induces a bounded operator on \(H\). Let \(I = (z_1, \dots , z_n)\) be the maximal ideal of \(A\) at the origin. If \(\dim H/IH < \infty\), then it is known that \(\dim H/I^kH < \infty\) for all \(k \in \mathbb N\), that the limit \[ e(H) = n! \lim_{k \to \infty} \frac{\dim H/I^kH}{k^n} \] exists and is an integer, called the Samuel multiplicity of \(H\); see \textit{X.~Fang} [Adv.\ Math.\ 186, No.~2, 411--437 (2004; Zbl 1070.47007)]. The author of the paper under review proves, for some certain classes of Hilbert modules, that the Fredholm index of a submodule is equal to its fibre dimension (an analytic notion) if and only if the Fredholm index of the quotient module is equal to its Samuel multiplicity (an algebraic notion).
reviewed by: Mohammad Sal Moslehian
Mathematics Subject Classification ID: 47A53
Mathematics Subject Classification ID: 46H99
Mathematics Subject Classification ID: 46M18
Mathematics Subject Classification ID: 47A13
Mathematics Subject Classification ID: 47B35
zbMATH DE Number: 5001039
zbMATH Keywords: Hilbert module
zbMATH Keywords: Fredholm index
zbMATH Keywords: fiber dimension
zbMATH Keywords: Samuel multiplicity
zbMATH Keywords: Fock space
zbMATH Keywords: Bergman space
MaRDI profile type: Publication
DOI: 10.4310/MRL.2005.V12.N6.A11
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description: scientific article; zbMATH DE number 5001039