دانلود کتاب Notes on the Brown-Douglas-Fillmore Theorem (Cambridge IISc Series)

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Cover
Notes on the Brown‒Douglas‒Fillmore Theorem
Series
Title
Copyright
Dedication
Contents
Preface
Overview
1 Spectral Theory for Hilbert Space Operators
1.1 Partial Isometries and Polar Decomposition
1.2 Compact and Fredholm Operators
1.3 Fredholm Index and Abstract Index
1.4 Schatten Classes
1.5 Isometries and von Neumann{Wold Decomposition
1.6 Toeplitz Operators with Continuous Symbols
1.7 Notes and Exercises
2 Ext(X) as a Semigroup with Identity
2.1 Essentially Normal Operators
2.2 Weyl{von Neumann{Berg{Sikonia Theorem
2.3 Extensions and Essentially Unitary Operators
2.4 Absorption Lemma
2.5 Weakly and Strongly Equivalent Extensions
2.6 Existence and Uniqueness of Trivial Class
2.7 Identity Element for Ext(X)
2.8 Notes and Exercises
3 Splitting and the Mayer–Vietoris Sequence
3.1 Splitting
3.2 Disjoint Sum of Extensions
3.3 First Splitting Lemma
3.4 Surjectivity of (iA;X)
3.5 Ext(A)→Ext(X)→Ext(X/A) is Exact
3.6 Mayer‒Vietoris Sequence
3.7 Notes and Exercises
4 Determination of Ext(X) as a Group for Planar Sets
4.1 Second Splitting Lemma
4.2 Projective Limits and Iterated Splitting Lemma
4.3 Ext(X) is a Group
4.4 γX is Injective
4.5 BDF Theorem and Its Consequences
4.6 Notes and Exercises
5 Applications to Operator Theory
5.1 Bergman Operators and Surjectivity of γX
5.2 Hyponormal Operators and m-Isometries
5.3 Essentially Normal Circular Operators
5.4 Essentially Homogeneous Operators
5.5 Essentially Reductive Quotient and Submodules
5.6 Notes and Exercises
Epilogue
Other Proofs
Properties of Ext(X)
The short exact sequence
Arveson\'s proof of \"Ext(X) is a group\"
Related Developments
Ext(A, B)
Homotopy invariance
K-theory
Open Problems
Essentially normal tuples
Essentially homogeneous tuples
Arveson{Douglas conjecture
Appendix A Point Set Topology
Urysohn\'s Lemma and Tietze Extension Theorem
Product Topology and and Tychono\'s Theorem
Alexandro{Hausdor Theorem and Totally Disconnected Metric Spaces
Quotient Topology
Appendix B Linear Analysis
Stone{Weierstrass Theorem
Hilbert spaces and Parseval\'s identity
Quotient spaces and C*-algebras
Open Mapping Theorem
Dual spaces and weak convergence
Spectrum of a Bounded Linear Operator
Functional Calculus for Bounded Linear Operators
Appendix C The Spectral Theorem
Spectral Theorem
Positive operators as directed set
Path-connectivity of the Group of Invertible Operators
Putnam{Fuglede Theorem
Outline of the Proof of the Spectral Theorem
The Multiplicity Function and the Spectral Theorem
References
Subject Index
Index of Symbols