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 | Agler J. Classical function theory, operator dilation theory, and machine computation on multiply-connected domains / Agler J., Harland J., Raphael B.J. - Providence: AMS, 2008. - 159 p. - (Memoirs of the American mathematical society; N 892). - ISSN 0065-9266; ISBN 0821840460
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Preface ....................................................... vii
Chapter 1. Generalizations of the Herglotz Representation
Theorem, von Neumann's Inequality and the Sz.-Nagy
Dilation Theorem to Multiply Connected Domains ....... 1
1.1. Introduction .................................... 1
1.2. Preliminaries ................................... 7
1.3. The First Herglotz Representation .............. 12
1.4. The Second Herglotz Representation ............. 21
1.5. The Third Herglotz Representation .............. 23
1.6. The Herglotz Representations and Operator
Theory ......................................... 28
1.7. An Application ................................. 33
Chapter 2. The Computational Generation of Counterexamples to
the Rational Dilation Conjecture .................... 61
2.1. Introduction ................................... 61
2.2. Mathematical Preliminaries ..................... 64
2.3. Analysis of the Dilation Condition for
Nonsingularly Hyperextremal Grammians .......... 72
2.4. Analysis of Dilation Extremal Grammians ........ 82
2.5. Algorithms ..................................... 89
2.6. A Computational Counterexample ................ 100
2.7. Plausibility Arguments ........................ 105
Chapter 3. Arbitrary Precision Computations of the Poisson
Kernel and Herglotz Kernels on Multiply-Connected
Circle Domains ..................................... 109
3.1. Introduction .................................. 109
3.2. Computation of the Functions .................. 112
3.3. Results ....................................... 119
Chapter 4. Schwartz Kernels on Multiply Connected Domains ..... 127
Appendix A. Convergence Results ............................... 139
Appendix B. Example Inner Product Computation ................. 155
Bibliography .................................................. 157
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