Samuel multiplicity and the structure of semi-Fredholm operators
type: Item
instance of: scholarly article
title: Samuel multiplicity and the structure of semi-Fredholm operators
zbMATH Open document ID: 1070.47007
author: Xiang Fang
published in: Advances in Mathematics
publication date: 2004-08-19T00:00:00Z
review text: In this paper, the shift Samual multiplicity, denoted \(s_{mul}(T)\), and the backward shift Samual multiplicity, denoted \(bs_{mul}(T)\), are introduced for any semi-Fredholm operator \(T\) on a separable complex Hilbert space \(H\): \[ s_{mul}(T)=\text{ lim}_{t\to \infty} ({\text{dim}(H/T^k(H))/k)}, \] \[ bs_{mul}(T)=\text{ lim}_{t\to \infty} (\text{dim} (\text{ker}(T^k)))/k). \] These numerical invariants refining the Fredholm index can be regarded as stabilized dimensions of the cokernel \(H/T(H)\) and kernel \(\ker (T)\), respectively. A geometric interpretation of \(s_{mul}(T)\) and \(bs_{mul}(T)\) leads to a local version of Apostol's triangular representation theorem and to Gohberg's punctured neighborhood theorem. Banach space operators are also considered.
Mathematics Subject Classification ID: 47A53
Mathematics Subject Classification ID: 47A13
Mathematics Subject Classification ID: 46H25
zbMATH DE Number: 2093017
zbMATH Keywords: semi-Fredholm operator
zbMATH Keywords: Fredholm index
zbMATH Keywords: Samuel multiplicity
zbMATH Keywords: Hilbert module
zbMATH Keywords: operator matrix
reviewed by: V. M. Deundjak
MaRDI profile type: Publication
full work available at URL: j.aim.2003.07.013
OpenAlex ID: W1984212629
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DOI: 10.1016/J.AIM.2003.07.013
description: scientific article; zbMATH DE number 2093017