An Introduction to Local Spectral Theory

by Laursen, Kjeld B.; Neuman, Michael M.

• ISBN13: 9780198523819
• ISBN10: 0198523815
• Format: Hardcover
• Copyright: 2000-06-01
• Publisher: Oxford University Press

Summary

Modern local spectral theory is built on the classical spectral theorem, a fundamental result in single-operator theory and Hilbert spaces. This book provides an in-depth introduction to the natural expansion of this fascinating topic of Banach space operator theory, whose pioneers includeDunford, Bishop, Foias, and others. Assuming only modest prerequisites of its readership, it gives complete coverage of the field, including the fundamental recent work by Albrecht and Eschmeier which provides the full duality theory for Banach space operators. It is highlighted by manycharacterizations of decomposable operators, and of other related, important classes of operators, as well as an in-depth study of their spectral properties, including identifications of distinguished parts, and results on permanence properties of spectra with respect to several types of similarity.Also found is a thorough and quite elementary treatment of the modern single- operator duality theory; this theory has many applications, both to general issues of classification and to such celebrated problems as the invariant subspace problems. A long chapter - almost a book in itself - is devotedto the use of local spectral theory in the study of spectral properties of multipliers and convolution operators. Another one describes its connections to automatic continuity theory. Written in a careful and detailed style, it contains numerous examples, many simplified proofs of classical results,and extensive references. It concludes with a list of interesting open problems, suitable for continued research.

Table of Contents

Decomposable operators	1	(98)
The evolution of axiomatic spectral theory	1	(6)
Characterizations of decomposability	7	(27)
Distinguished parts of the spectrum	34	(4)
Super-decomposable operators	38	(16)
Generalized scalar operators	54	(22)
Local spectral properties of isometries and shifts	76	(14)
Growth conditions and Bishop's property (β)	90	(7)
Summary of local spectral properties	97	(1)
Functional models, duality theory, and invariant subspaces	98	(99)
Tensor products and analytic functions	99	(13)
The three-space lemmas for properties (β) and (δ)	112	(8)
Interlude on Bergman and Sobolev-type spaces	120	(14)
Decomposable extensions and liftings	134	(15)
The duality between property (β) and property (δ)	149	(25)
Existence of invariant subspaces	174	(23)
The spectrum and spectral inclusions	197	(102)
The Kato spectrum	199	(12)
The Allan--Leiterer theory	211	(9)
Spectral subspaces	220	(23)
Local spectral inclusions and permanence properties	243	(12)
Perturbed spectral inclusions	255	(15)
Local spectral properties of commutators	270	(14)
Fredholm theory and essential spectra	284	(15)
Local spectral theory for multipliers	299	(167)
Multipliers on commutative Banach algebras	300	(18)
Permanence properties of decomposable multipliers	318	(11)
Regular algebras and the hull-kernel topology	329	(19)
Decomposable multiplication operators	348	(14)
Multipliers in Mo(A) and Moo(A)	362	(6)
Natural spectrum and symmetric algebras	368	(9)
Natural local spectra and Tauberian algebras	377	(9)
Banach algebras with scattered or discrete spectra	386	(10)
Compact and Riesz multipliers	396	(9)
Multipliers with closed range	405	(16)
Decomposable convolution operators	421	(13)
Arveson spectrum and spectral mapping theorems	434	(18)
Banach algebras of vector-valued functions	452	(14)
Connections to automatic continuity	466	(67)
Some basic principles of automatic continuity	467	(9)
Translation-invariance and causality	476	(7)
Singularity points for mappings of local type	483	(11)
Intertwining linear transformations	494	(11)
Non-analytic functional calculi	505	(15)
Homomorphisms and derivations on Co(Ω.A)	520	(13)
Open problems	533	(7)
Operator theory	533	(3)
Banach algebras and multipliers	536	(3)
Automatic continuity	539	(1)
Appendix	540	(9)
A.1 Basic functional analysis	540	(4)
A.2 Banach algebras	544	(3)
A.3 Vector-valued analytic functions	547	(2)
Bibliography	549	(26)
Index of notation	575	(2)
Index	577