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El. knyga: Notes on the Brown-Douglas-Fillmore Theorem

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(Indian Institute of Science, Bangalore), (Indian Institute of Technology, Kanpur)
  • Formatas: PDF+DRM
  • Serija: Cambridge IISc Series
  • Išleidimo metai: 07-Oct-2021
  • Leidėjas: Cambridge University Press
  • Kalba: eng
  • ISBN-13: 9781009032452
Kitos knygos pagal šią temą:
Notes on the Brown-Douglas-Fillmore TheoremDidesnis paveikslėlis
  • Formatas: PDF+DRM
  • Serija: Cambridge IISc Series
  • Išleidimo metai: 07-Oct-2021
  • Leidėjas: Cambridge University Press
  • Kalba: eng
  • ISBN-13: 9781009032452
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Suitable for both postgraduate students and researchers in the field of operator theory, this book is an excellent resource providing the complete proof of the Brown-Douglas-Fillmore theorem. The book starts with a rapid introduction to the standard preparatory material in basic operator theory taught at the first year graduate level course. To quickly get to the main points of the proof of the theorem, several topics that aid in the understanding of the proof are included in the appendices. These topics serve the purpose of providing familiarity with a large variety of tools used in the proof and adds to the flexibility of reading them independently.

An excellent resource for both postgraduate students and researchers in the field of mathematics, this book gives a complete proof of the Brown-Douglas-Fillmore theorem along with a number of its applications and several open problems.

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The book discusses the complete proof of the Brown-Douglas-Fillmore theorem along with a number of its applications.
Preface ix
Overview 3(4)
1 Spectral Theory for Hilbert Space Operators
7(50)
1.1 Partial Isometries and Polar Decomposition
7(2)
1.2 Compact and Fredholm Operators
9(10)
1.3 Fredholm Index and Abstract Index
19(15)
1.4 Schatten Classes
34(3)
1.5 Isometries and von Neumann-Wold Decomposition
37(6)
1.6 Toeplitz Operators with Continuous Symbols
43(4)
1.7 Notes and Exercises
47(10)
2 Ext(X) as a Semigroup with Identity
57(34)
2.1 Essentially Normal Operators
57(3)
2.2 Weyl-von Neumann-Berg-Sikonia Theorem
60(6)
2.3 Extensions and Essentially Unitary Operators
66(4)
2.4 Absorption Lemma
70(3)
2.5 Weakly and Strongly Equivalent Extensions
73(3)
2.6 Existence and Uniqueness of Trivial Class
76(6)
2.7 Identity Element for Ext(X)
82(4)
2.8 Notes and Exercises
86(5)
3 Splitting and the Mayer-Vietoris Sequence
91(26)
3.1 Splitting
91(3)
3.2 Disjoint Sum of Extensions
94(3)
3.3 First Splitting Lemma
97(3)
3.4 Surjectivity of
100(8)
3.5 Ext(.A) → Ext(X) → Ext(X/A) is Exact
108(4)
3.6 Mayer-Vietoris Sequence
112(2)
3.7 Notes and Exercises
114(3)
4 Determination of Ext(X) as a Group for Planar Sets
117(30)
4.1 Second Splitting Lemma
117(7)
4.2 Projective Limits and Iterated Splitting Lemma
124(6)
4.3 Ext(X) is a Group
130(3)
4.4 γx is Injective
133(7)
4.5 BDF Theorem and Its Consequences
140(4)
4.6 Notes and Exercises
144(3)
5 Applications to Operator Theory
147(32)
5.1 Bergman Operators and Surjectivity of γx
147(5)
5.2 Hyponormal Operators and m-Isometries
152(5)
5.3 Essentially Normal Circular Operators
157(2)
5.4 Essentially Homogeneous Operators
159(5)
5.5 Essentially Reductive Quotient and Submodules
164(11)
5.6 Notes and Exercises
175(4)
Epilogue
179(14)
Other Proofs
179(1)
Properties of Ext(X)
180(1)
The short exact sequence
180(1)
Arveson's proof of "Ext(Z) is a group"
181(1)
Related Developments
182(1)
Ext(A,B)
182(1)
Homotopy invariance
183(2)
K-theory
185(2)
Open Problems
187(1)
Essentially normal tuples
187(2)
Essentially homogeneous tuples
189(1)
Arveson-Douglas conjecture
190(3)
Appendix A Point Set Topology 193(8)
Appendix B Linear Analysis 201(16)
Appendix C The Spectral Theorem 217(16)
References 233(8)
Subject Index 241(4)
Index of Symbols 245
Sameer Chavan is Professor at the Department of Mathematics and Statistics, Indian Institute of Technology Kanpur. He works on function-theoretic and graph-theoretic operator theory. He was P. K. Kelkar Fellow for the period 20172020. Gadadhar Misra is Professor at the Department of Mathematics, Indian Institute of Science Bangalore. He works in complex geometry and operator theory. He was awarded the Shanti Swarup Bhatnagar Prize in 2001. He is a fellow of all the three science academies in India and is a J C Bose National Fellow.
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