Banach Space Complexes

ISBN-13: 9780792336303 / Angielski / Twarda / 1995 / 213 str.

Calin-Grigore Ambrozie; C. G. Ambrozie; Florian-Horia Vasilescu

Opis

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.

Spis treści

Introduction. I: Preliminaries. 1. Algebraic prerequisites. 2. Algebraic Fredholm pairs. 3. Paraclosed linear transformations. 4. Homogeneous operators. 5. Linear and homogeneous projections and liftings. 6. The gap between two closed subspaces. 7. Linear operators with closed range, and finite extensions. 8. Metric relations and duality. 9. Operators in quotient Banach spaces. 10. References and comments. II: Semi-Fredholm complexes. 1. Semi-Fredholm operators. 2. Semi-Fredholm complexes. 3. Essential complexes. 4. Fredholm pairs. 5. Other continuous invariants. 6. References and comments. III: Related topics. 1. Joint spectra and perturbations. 2. Spectral interpolation and perturbations. 3. Versions of Poincaré's and Grothendieck's lemmas. 4. Differentiable families of partial differential operators. 5. References and comments. Subject index. Notations. Bibliography.