Table of contents for Spectral properties of noncommuting operators / Brian Jefferies.

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1   Introduction  ........... ...... ............................   1
2   Weyl Calculus ..........................................   . 13
2.1 Background ............................................. 13
2.2 Operators of Paley-Wiener Types .......................... 16
2.3 The Joint Spectrum ..................................... 21
3   Clifford Analysis .................  ...................... ..27
3.1  Clifford Algebras ............ ..  ............  . .  .  27
3.2 Banach Modules ...................................... 29
3.3 Cauchy Formula ............... ................... ...   30
3.4 Vector Valued Functions...................               32
3.5 Monogenic Expansions ............................. 33
3.6 Monogenic Representation of Distributions ................. 35
3.7 Plane Wave Decomposition ...............      ........... 35
3.8 Approximation ............    .....................   . 36
4   Functional Calculus for Noncommuting Operators ......... 39
4.1 The Weyl Calculus and the Cauchy Kernel ................. 39
4.2 The Joint Spectrum and the Cauchy Kernel ................ 44
4.3 The Monogenic Functional Calculus ....................... 52
4.4  Spectral Decomposition  ....................  . . . . . .  .  60
5   The Joint Spectrum of Matrices .......................... 67
5.1 Nelson's Formula for Hermitian Matrices .................. 67
5.2 Exponential Bounds for Matrices    ........................ 73
5.2.1  Perturbation  ......................................  74
5.2.2 The Exponential Bound ...................... . ..  77
5.3 The Joint Spectrum of Pairs of Hermitian Matrices ......... 80
5.3.1 The Numerical Range of Matrices .................. 81
5.3.2 Examples .................................... .88
5.3.3 The Joint Spectrum of Two Hermitian Matrices ....... 88



5.4 Simultaneously Triangularisable Matrices ................... 104
5.4.1  Disintegration of Measures  ................................ 106
5.4.2 The Image of Simplicial Measure ................... 107
5.4.3 Joint Spectrum of Triangularisable Matrices .......... 110
5.5 Systems of Matrices .    ........................................111
6   The Monogenic Calculus for Sectorial Operators .......... 123
6.1 The H°-Functional Calculus for a Single Operator .......... 124
6.2  The Cauchy Kernel for n Sectorial Operators ............... 126
6.3 Monogenic and Holomorphic Functions in Sectors ........... 130
6.3.1  Joint Spectral Theory in the Algebra C(n) ............ 130
6.3.2 Plane Wave Decompositions ........................ 135
6.3.3  Bounded Monogenic Functions in a Sector ............ 137
6.4 Bounded Holomorphic Functions in Sectors ................. 140
6.4.1  Sectors in C  ...................................... 140
6.4.2  Fourier Analysis in Sectors ................. . ..... 142
6.5 The Monogenic Calculus for n Sectorial Operators .......... 153
7   Feynman's Operational Calculus                ......... ...........................157
7.1 Operants for the Weyl Calculus ........................... 157
7.2 Feynman's gt-Operational Calculus for n Operators .......... 160
7.3 The pA-Monogenic Calculus for n Operators .................. 165
References  .............. ...            ................................ 173
Index  of Notation  ............................................. 181
Index  .......................................................... .183