Finite rank modifications and generalized inverses of Fredholm operators

AbstractThe Laffey–West theorem concerning finite rank perturbations of bounded Fredholm operators is extended to closed densely defined operators on Banach Spaces.

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A Trace Formula for the Index of B-Fredholm Operators

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091518000147 ◽

2018 ◽

Vol 61 (4) ◽

pp. 1063-1068 ◽

Cited By ~ 2

Author(s):

Mohammed Berkani

Keyword(s):

Banach Space ◽

Banach Algebra ◽

Fredholm Operator ◽

Trace Formula ◽

Finite Rank ◽

Toeplitz Operators ◽

Drazin Inverse ◽

Trace Function ◽

Fredholm Operators ◽

The Ideal

AbstractIn this paper we define B-Fredholm elements in a Banach algebraAmodulo an idealJofA. When a trace function is given on the idealJ, it generates an index for B-Fredholm elements. In the case of a B-Fredholm operatorTacting on a Banach space, we prove that its usual index ind(T) is equal to the trace of the commutator [T, T0], whereT0is a Drazin inverse ofTmodulo the ideal of finite rank operators, extending Fedosov's trace formula for Fredholm operators (see Böttcher and Silbermann [Analysis of Toeplitz operators, 2nd edn (Springer, 2006)]. In the case of a primitive Banach algebra, we prove a punctured neighbourhood theorem for the index.

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Some singular values and unitarily invariant norm inequalities concerning generalized inverses

Filomat ◽

10.2298/fil0701099m ◽

2007 ◽

Vol 21 (1) ◽

pp. 99-111

Author(s):

P.J. Maher

Keyword(s):

Finite Rank ◽

Singular Values ◽

Generalized Inverses ◽

Unitarily Invariant Norm ◽

Norm Inequalities ◽

Von Neumann ◽

Finite Rank Operators ◽

Invariant Norm

We sharpen and extend inequalities concerning generalized inverses previously obtained for the von Neumann-Schatten, and supremum, norms. We sharpen those inequalities to obtain corresponding inequalities for singular values Si(?) for i=1,2?; and we extend those inequalities, for finite rank operators, to inequalities for an arbitrary unitarily invariant norm.

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Canonical holomorphic sections of determinant line bundles

Journal für die reine und angewandte Mathematik (Crelles Journal) ◽

10.1515/crelle-2015-0114 ◽

2019 ◽

Vol 2019 (746) ◽

pp. 67-116 ◽

Cited By ~ 1

Author(s):

Jens Kaad ◽

Ryszard Nest

Keyword(s):

Finite Rank ◽

Main Tool ◽

Line Bundles ◽

Fredholm Operators ◽

Homology Groups ◽

Invariance Properties ◽

Holomorphic Sections ◽

Finite Rank Perturbation ◽

Determinant Line Bundles ◽

Fredholm Complexes

Abstract We investigate the analytic properties of torsion isomorphisms (determinants) of mapping cone triangles of Fredholm complexes. Our main tool is a generalization to Fredholm complexes of the perturbation isomorphisms constructed by R. Carey and J. Pincus for Fredholm operators. A perturbation isomorphism is a canonical isomorphism of determinants of homology groups associated to a finite rank perturbation of Fredholm complexes. The perturbation isomorphisms allow us to establish the invariance properties of the torsion isomorphisms under finite rank perturbations. We then show that the perturbation isomorphisms provide a holomorphic structure on the determinant lines over the space of Fredholm complexes. Finally, we establish that the torsion isomorphisms and the perturbation isomorphisms provide holomorphic sections of certain determinant line bundles.

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