Reciprocity for Fredholm operators (Lefschetz numbers/Steinberg symbols/holomorphic chains/local index)
type: Item
instance of: scholarly article
title: Reciprocity for Fredholm operators (Lefschetz numbers/Steinberg symbols/holomorphic chains/local index)
zbMATH Open document ID: 0622.47009
DOI: 10.1007/BF01204625
author: Joel D. Pincus
author: Richard W. Carey
published in: Integral Equations and Operator Theory
publication date: 1986-01-01T00:00:00Z
review text: The difference between the joint spectrum and the essential joint spectrum of a pair of commuting bounded operators acting on a Hilbert space consists of connected open subsets of \({\mathbb{C}}^ 2\), analytic varieties of complex dimension 1 and isolated points. The authors of the present paper have developed a refined index theory which assigns certain multiplicities even to the above mentioned low dimensional parts of the essential resolvent set, cf. \textit{R. W. Carey} and \textit{J. D. Pincus}: Principal currents, ibid. 8, 614-640 (1985). The present paper fits into the same program. To a pair of Fredholm operators A and B is associated the Fredholm number \[ \psi (B;A)=\det (B| Ker A)/\det (B| Co\ker A). \] The relationship between \(\psi\) (B;A) and \(\psi\) (A;B) is studied from the point of view of algebraic K- theory. Then it is proved that the Lefschetz number preserves the analytic dependence on parameters. With the aid of these tools, the authors investigate the properties of their refined index. Some interesting connections with other areas of mathematics promise a continuation and an interesting evolution of the ideas of Carey and Pincus. However, the paper is intended to mature readers.
reviewed by: Mihai Putinar
Mathematics Subject Classification ID: 47A53
Mathematics Subject Classification ID: 18F25
Mathematics Subject Classification ID: 47A10
zbMATH DE Number: 4009127
zbMATH Keywords: analytic chain
zbMATH Keywords: essential joint spectrum of a pair of commuting bounded operators acting on a Hilbert space
zbMATH Keywords: index theory
zbMATH Keywords: Fredholm operators
zbMATH Keywords: Fredholm number
zbMATH Keywords: algebraic K-theory
zbMATH Keywords: Lefschetz number
zbMATH Keywords: analytic dependence on parameters
MaRDI profile type: Publication
cites work: Q3922789
cites work: Principal currents
cites work: Q3728652
cites work: Fredholm and Invertible n-Tuples of Operators. The Deformation Problem
cites work: Q5526485
cites work: Introduction to Algebraic K-Theory. (AM-72)
cites work: The spectrum of difference operators and algebraic curves
cites work: Q4741047
cites work: On the removal of singularities of analytic sets
cites work: On holomorphic families of subspaces of a Banach space
cites work: Q3921379
cites work: A joint spectrum for several commuting operators
cites work: The analytic-functional calculus for several commuting operators
cites work: Q3933478
cites work: Q5682591
description: scientific article; zbMATH DE number 4009127