Recent Trends in Toeplitz and Pseudodifferential Operators (eBook)

Table of Contents 6
The Life and Work of Nikolai Vasilevski 9
On the Structure of the Eigenvectors of Large Hermitian Toeplitz Band Matrices 23
1. Introduction and main results 23
2. The first column of the adjugate matrix 29
3. The main terms of the first column 32
4. The asymptotics of the eigenvectors 37
5. Symmetric matrices 39
6. Numerical results 42
References 43
Complete Quasi-wandering Sets and Kernels of Functional Operators 45
1. Introduction 45
2. The kernel space of the operator Ua 47
References 50
Lions’ Lemma, Korn’s Inequalities and the Lam´e Operator on Hypersurfaces 51
Introduction 51
1. Sobolev spaces and Bessel potential operators 56
2. Lions’ Lemma and Korn’s inequalities 59
3. Killing’s vector fields and the unique continuation from the boundary 63
4. A local fundamental solution to the Lame equation 70
5. BVPs for the Lame equation and Green’s formulae 73
6. The Dirichlet BVP for the Lame equation 76
7. The Neumann BVP for the Lame equation 78
References 82
On the Bergman Theory for Solenoidal and Irrotational Vector Fields, I: General Theory 86
1. Introduction 86
1.1. 86
1.2. 87
1.3. 87
2. Solenoidal and irrotational vector fields: main results 88
2.1. 88
2.2. The SI-Bergman space and the SI-Bergman kernel 88
2.3. SI-Bergman projection 90
2.4. Decomposition of ˆ L2 90
2.5. M¨obius transformations in R3 in vectorial language 91
2.6. R-linear spaces of vector fields and M¨obius transformations on R3 92
3. The Bergman theory for Moisil-Theodoresco hyperholomorphy 95
3.1. Preliminaries 95
3.2. Mobius transformations in R3 in quaternionic terms 96
3.3. Mobius transformations and hyperholomorphy 97
3.4. Some properties of quaternionic Hilbert spaces 99
3.5. The hyperholomorphic Bergman space 100
3.6. Mobius transformations in R3 and function spaces 104
3.7. Decomposition of the space L2(O,H) 106
4. The SI- and hyperholomorphic Bergman spaces 107
4.1. Preliminaries 107
4.2. Proofs of the statements of Section 2 109
References 112
Weighted Estimates of Generalized Potentials in Variable Exponent Lebesgue Spaces on Homogeneous Spaces 114
1. Introduction 114
2. Formulation of the main result 116
3. Auxiliary estimates 118
4. Main lemma 121
5. Proof of the main result 125
References 127
Wiener-Hopf Operators with Oscillating Symbols on Weighted Lebesgue Spaces 130
1. Introduction 130
2. Auxiliary results 133
2.1. Slowly oscillating Fourier multipliers on Lp(R,w) 133
2.2. Limit operators and the fiber M8([SO, SAP]) 134
2.3. Wiener-Hopf operators with almost periodic symbols 136
3. Geometric mean for matrix functions in [APp]N×N 137
4. Necessary Fredholm conditions 140
5. Regularizers of Wiener-Hopf operators with symbols in SAPp,w 142
6. Wiener-Hopf operators with symbols in [SOp,w, SAPp,w] 143
7. Matrix case 149
References 151
On the Ill-posed Hyperbolic Systems with a Multiplicity Change Point of Not Less Than the Third Order 153
1. Main results 153
2. Ramifications of trajectories 159
3. Proof of the theorem 165
4. Addition 176
References 179
On C*-Algebras of Super Toeplitz Operators with Radial Symbols 181
1. Introduction 181
2. Super Toeplitz operators on the unit disk 182
3. Action of the super circle on the super disk 185
4. Radial functions: S1|1-invariant functions 187
5. Super Toeplitz operators with radial symbols 189
6. A generalization of the Toeplitz algebra with radial symbol (classical) 193
References 193
Universality of Some C*-Algebra Generated by a Unitary and a Self-adjoint Idempotent 195
1. Introduction 195
2. The main result and its proof 196
References 199
Commutative Algebras of Toeplitz Operators and Lagrangian Foliations 200
1. Introduction 200
2. Preliminaries on symplectic geometry and foliations 202
3. Berezin’s correspondence principle for bounded domains 203
4. Proofs of the main results 204
References 205
Exponential Estimates of Eigenfunctions of Matrix Schr¨odinger and Dirac Operators 207
1. Introduction 207
2. Exponential estimates of solutions of systems of partial differential equations 208
2.1. Essential spectrum 208
2.2. Exponential estimates 210
3. Schrödinger operators with matrix potentials 211
3.1. Essential spectrum 211
3.2. Exponential estimates of eigenfunctions of the discrete spectrum 213
4. Dirac operators 215
4.1. Essential spectrum of Dirac operators 215
4.2. Exponential estimates of eigenfunctions of the Dirac operator 217
References 218
The Laplace-Beltrami Operator on a Rotationally Symmetric Surface 221
Introduction 221
1. Geometry 223
2. Laplace-Beltrami operator 224
3. Weighted spaces 225
4. Resolvent 228
5. Fredholm theory 230
6. Asymptotics 231
7. Index formula 233
References 234
On the Structure of Operators with Automorphic Symbols 236
1. Introduction 236
2. Operators with automorphic symbols 237
3. Some automorphic operators 239
4. The distribution T8 242
5. Other symbolic calculi 243
References 251
Hilbert Bundles and Flat Connexions over Hermitian Symmetric Domains 252
0. Introduction 252
1. The canonical connexion on a Hilbert subbundle 253
2. The Fock bundle of entire functions 257
3. Fock bundles for Jordan algebra representations 264
4. The Jordan theoretic framework 269
References 274