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Banach Space Complexes

By: C.-G. Ambrozie, Florian-Horia Vasilescu

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Hardcover | 1 August 1995

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224 Pages

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25.4 x 17.15 x 1.91

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The aim of this work is to initiate a systematic study of those properties of Banach space complexes that are stable under certain perturbations. A Banach space complex is essentially an object of the form 1 op-l oP +1 ... --+ XP- --+ XP --+ XP --+ ... , where p runs a finite or infiniteinterval ofintegers, XP are Banach spaces, and oP : Xp ..... Xp+1 are continuous linear operators such that OPOp-1 = 0 for all indices p. In particular, every continuous linear operator S : X ..... Y, where X, Yare Banach spaces, may be regarded as a complex: O ..... X ~ Y ..... O. The already existing Fredholm theory for linear operators suggested the possibility to extend its concepts and methods to the study of Banach space complexes. The basic stability properties valid for (semi-) Fredholm operators have their counterparts in the more general context of Banach space complexes. We have in mind especially the stability of the index (i.e., the extended Euler characteristic) under small or compact perturbations, but other related stability results can also be successfully extended. Banach (or Hilbert) space complexes have penetrated the functional analysis from at least two apparently disjoint directions. A first direction is related to the multivariable spectral theory in the sense of J. L.

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Non-Fiction Mathematics Calculus & Mathematical Analysis Functional Analysis & Transforms Topology Differential Calculus & Equations Gardening Complex Analysis

Introduction	p. 1
Preliminaries	p. 3
Algebraic prerequisites	p. 3
Algebraic Fredholm pairs	p. 8
Paraclosed linear transformations	p. 18
Homogeneous operators	p. 25
Linear and homogeneous projections and liftings	p. 33
The gap between two closed subspaces	p. 39
Linear operators with closed range, and finite extensions	p. 48
Metric relations and duality	p. 53
Operators in quotient Banach spaces	p. 64
Semi-Fredholm complexes	p. 69
Semi-Fredholm operators	p. 69
Semi-Fredholm complexes	p. 82
Essential complexes	p. 94
Fredholm pairs	p. 119
Other continuous invariants	p. 137
Related topics	p. 153
Joint spectra and perturbations	p. 153
Spectral interpolation and perturbations	p. 173
Versions of Poincare's and Grothendieck's lemmas	p. 176
Differentiable families of partial differential operators	p. 187
Bibliography	p. 197
Subject index	p. 203
Notations	p. 204
Table of Contents provided by Blackwell. All Rights Reserved.

Banach Space Complexes

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