Operator Theory and Indefinite Inner Product Spaces (eBook)

Presented on the Occasion of the Retirement of Heinz Langer in the Colloquium on Operator Theory, Vienna, March 2004

Matthias Langer, Annemarie Luger, Harald Woracek (Herausgeber)

eBook Download: PDF

2006 | 2006
XXXII, 381 Seiten
Springer Basel (Verlag)
978-3-7643-7516-4 (ISBN)

Infos

A colloquium on operator theory was held in Vienna, Austria, in March 2004, on the occasion of the retirement of Heinz Langer, a leading expert in operator theory and indefinite inner product spaces. The book contains fifteen refereed articles reporting on recent and original results in various areas of operator theory, all of them related with the work of Heinz Langer. The topics range from abstract spectral theory in Krein spaces to more concrete applications, such as boundary value problems, the study of orthogonal functions, or moment problems. The book closes with a historical survey paper.

Inhalt

Table of Contents 6
Preface 8
Laudation 9
Speech of Heinz Langer 15
Programme of the "Colloquium on Operator Theory” 18
Thursday, 4 March 18
Friday, 5 March 19
Saturday, 6 March, morning 20
Saturday, 6 March, afternoon 21
Participants of the “Colloquium on Operator Theory” 22
Bibliography of Heinz Langer 28
General Solution of the Stieltjes Truncated Matrix Moment Problem 31
1. Introduction 31
2. Reduction to the extension theory problem 33
3. Solution in the completely degenerate case 36
4. Canonical solutions 40
5. Non-canonical solutions 50
References 51
Q-functions of Quasi-selfadjoint Contractions 53
1. Introduction 53
2. Preliminaries 55
3. qsc-extensions of closed symmetric contractions 60
4. Q-functions of qsc-operators 65
5. The resolvent formula for qsc-extensions 70
6. A reproducing kernel space model for Q-functions of quasi-selfadjoint contractions 73
7. Characteristic properties of Q-functions of quasi-selfadjoint contractions 77
8. Linear fractional transformations of the class 80
References 81
A Class of Abstract Boundary Value Problems with Locally De.nitizable Functions in the Boundary Condition 85
1. Introduction 85
2. Locally de.nitizable selfadjoint relations and locally definitizable functions 86
3. Boundary value problems with locally definitizable functions in the boundary condition 91
References 102
Riesz Bases of Root Vectors of Inde.nite Sturm-Liouville Problems with Eigenparameter Dependent Boundary Conditions, I 104
1. Introduction 104
2. Operators associated with the eigenvalue problem 107
3. Smooth connection and associated operator 109
4. Condition at 0 and associated operator 114
5. Conditions at –1 and 1, and associated operators 117
6. Riesz basis of root vectors 119
References 123
Minimal Models for Nn-functions 125
1. Introduction 125
2. Characterization of the class 129
3. Reproducing kernel Pontryagin spaces and canonical models 132
4. Minimal models in the space 134
5. A decomposition of 141
6. Minimal models in the space 146
7. Block operator matrix models in the space 155
8. Examples 156
References 160
Generalized Friedrichs Extensions Associated with Interface Conditions for Sturm-Liouville Operators 163
1. Introduction 163
2. Preliminaries 164
3. Sturm-Liouville expressions and interface conditions 167
4. Generalized Friedrichs extensions and interface conditions 169
References 172
Spectral Properties of Operator Polynomials with Nonnegative Coe.cients 174
1. Introduction 174
2. Ordered Banach algebras 175
3. The spectral radius of 177
as a function on the nonnegative axis 177
4. Factorization in ordered decomposing Banach algebras 179
5. Factorization of polynomials with nonnegative coe.cients 182
6. Operator polynomials with nonnegative coe.cients 185
References 188
Szegö Pairs of Orthogonal Rational Matrixvalued Functions on the Unit Circle 190
0. Introduction 190
1. Notation and preliminaries 192
2. Szeg. o pairs of orthonormal systems 196
3. Szeg. o pairs of rational matrix-valued functions 203
4. Integral representations of Szeg. o parameters 208
References 215
Singularities of Generalized Strings 217
1. Introduction 217
2. Some classes of functions and their interrelation 221
3. The square root transformation 231
4. Chains of matrix functions 242
5. Transformation of matrix chains 246
6. Evolution of singularities 259
7. On generalized strings 265
References 272
Orthogonal Polynomials on the Unit Circle with Respect to a Rational Weight Function 275
References 282
Bi-dimensional Moment Problems and Regular Dilations 283
1. Introduction 283
2. Notation and preliminaries 285
3. General solutions 287
4. Isometric solutions 294
5. Unitary solutions 297
References 299
Isospectral Vibrating Systems, Part 2: Structure Preserving Transformations 301
1. Introduction 301
2. Structure preserving transformations 303
3. Standard pairs and triples 305
4. Constructing SPE and SPS transformations 307
5. Algorithms 310
6. Systems with symmetries 311
7. Monic Hermitian systems with 314
8. Isospectral families of Hermitian, or real symmetric systems 317
9. Systems in reduced form 319
10. Conclusions 323
References 323
A Functional Description for the Commutative WJ.-algebras of the D+. -class 325
Introduction 325
1. Preliminaries 326
2. Unbounded elements in Banach spaces 332
3. A function model for a 338
symmetric family of the class 338
4. Functional representation of commutative 341
algebras 341
of the class 341
with a unique singular peculiarity 341
5. 352
algebras with an arbitrary number of peculiarities 352
6. Closing remarks 358
References 359
On Normal Extensions of Unbounded Operators: IV. A Matrix Construction 362
Operator matrices versus matrix operators 363
Andˆ o’s construction in unbounded circumstances su.ciency
Andˆ o’s construction in unbounded circumstances necessity
References 375
Directing Mappings in Kre . in Spaces 376
1. Introduction 376
2. De.nitizable operators 377
3. Preliminaries on integration 378
4. Directing mappings 380
5. The main theorem 385
References 388
The Extension Problem for Positive Definite Functions. A Short Historical Survey 389
1. Introduction 389
2. The emergence of the concept of positive de.nite functions 390
3. Results on the existence of extensions having certain properties 393
4. Description of all extensions 397
References 400

Produkt-Details

• Erscheint lt. Verlag: 16.6.2006
• Reihe/Serie: Operator Theory: Advances and Applications
• Reihe/Serie: Operator Theory: Advances and Applications
• Zusatzinfo: XXXII, 381 p.
• Verlagsort: Basel
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik
• Themenwelt: Technik
• Schlagworte: Boundary value problem • differential equation • indefinite inner product spaces • Matrix • matrix functions • matrix theory • Minimum • operator theory • orthogonal polynomials • Sturm-Liouville problem
• ISBN-10: 3-7643-7516-7 / 3764375167
• ISBN-13: 978-3-7643-7516-4 / 9783764375164