Operator Theory

1. Front Matter

   Pages i-xix

   PDF

2. Reproducing Kernel Hilbert Spaces

   1. Front Matter

      Pages 1-1

      PDF

   2. The Reproducing Kernel Property and Its Space: The Basics

      • Franciszek Hugon Szafraniec

      Pages 3-30

   3. The Reproducing Kernel Property and Its Space: More or Less Standard Examples of Applications

      • Franciszek Hugon Szafraniec

      Pages 31-58

   4. The Use of Kernel Functions in Solving the Pick Interpolation Problem

      • Jim Agler, John E. McCarthy

      Pages 59-71

   5. Bergman Kernel in Complex Analysis

      • Łukasz Kosiński, Włodzimierz Zwonek

      Pages 73-86

   6. Sampling Theory and Reproducing Kernel Hilbert Spaces

      • Antonio G. García

      Pages 87-110

   7. Reproducing Kernels in Coherent States, Wavelets, and Quantization

      • Syed Twareque Ali

      Pages 111-125

   8. Geometric Perspectives on Reproducing Kernels

      • Daniel Beltiţǎ, José E. Galé

      Pages 127-148

3. Indefinite Inner Product Spaces

   1. Front Matter

      Pages 149-150

      PDF

   2. Multi-valued Operators/Linear Relations Between Kreĭn Spaces

      • HendrikLuit Wietsma

      Pages 151-163

   3. Symmetric and Isometric Relations

      • Hendrik Luit Wietsma

      Pages 165-182

   4. Boundary Triplets, Weyl Functions, and the Kreĭn Formula

      • Vladimir Derkach

      Pages 183-218

   5. Contractions and the Commutant Lifting Theorem in Kreĭn Spaces

      • Michael Dritschel

      Pages 219-239

   6. Locally Definitizable Operators: The Local Structure of the Spectrum

      • Carsten Trunk

      Pages 241-259

   7. Schur Analysis in an Indefinite Setting

      • Aad Dijksma

      Pages 261-310

   8. Reproducing Kernel Kreĭn Spaces

      • Aurelian Gheondea

      Pages 311-343

   9. Generalized Nevanlinna Functions: Operator Representations, Asymptotic Behavior

      • Annemarie Luger

      Pages 345-371

   10. Indefinite Hamiltonians

       • Michael Kaltenbäck

       Pages 373-394

   11. The Critical Point Infinity Associated with Indefinite Sturm–Liouville Problems

       • Andreas Fleige

       Pages 395-429

A one-sentence definition of operator theory could be: The study of (linear) continuous operations between topological vector spaces, these being in general (but not exclusively) Fréchet, Banach, or Hilbert spaces (or their duals). Operator theory is thus a very wide field, with numerous facets, both applied and theoretical. There are deep connections with complex analysis, functional analysis, mathematical physics, and electrical engineering, to name a few. Fascinating new applications and directions regularly appear, such as operator spaces, free probability, and applications to Clifford analysis. In our choice of the sections, we tried to reflect this diversity. This is a dynamic ongoing project, and more sections are planned, to complete the picture. We hope you enjoy the reading, and profit from this endeavor.

• Schur analysis
• de Branges spaces
• indefinite inner product spaces
• infinite dimensional analysis
• noncommutative analysis
• reproducing kernel Hilbert spaces

• Earl Katz Chair in Algebraic System Theory, Department of Mathematics, Ben-Gurion University of the Negev, Be’er Sheva, Israel

  Daniel Alpay

Editorial Board:

• Daniel Alpay (Editor-in-Chief), Earl Katz Chair in Algebraic System   Theory, Department of Mathematics, Ben-Gurion University of the Negev, Be’er Sheva, Israel
  
• Joseph A. Ball, Department of Mathematics, Virginia Tech, Blacksburg, VA, USA
  
• Anton Baranov, Department of Mathematics and Mechanics, St. Petersburg State University, Pedrodvorets, Russia
  
• Fabrizio Colombo, Dipartimento di Matematica, Politecnico di Milano, Milano, Italy
  
• Palle E.T. Jorgensen, Department of Mathematics, The University of Iowa, Iowa City, IA, USA
  
• Matthias Langer, Department of Mathematics and Statistics, University of Strathclyde, Glasgow, Scotland, UK 
  
• Mamadou Mboup, Université de Reims Champagne Ardenne, CReSTIC - UFR des Sciences Exactes et Naturelles Moulin de la Housse, Reims, France
  
• Irene Sabadini, Dipartimento diMatematica, Politecnico di Milano, Milano, Italy
  
• Michael Shapiro, Departamento de Matemáticas, Escuela Superior de Física y Matemáticas, del Instituto Politécnico Nacional, Mexico City, Mexico
  
• Franciszek Hugon Szafraniec,Instytut Matematyki, Uniwersytet Jagiellónski, Kraków, Poland
  
• Harald Woracek, Institut for Analysis and Scientific Computing, Vienna University of Technology, Vienna, Austria
  

Prof. Daniel Alpay is a faculty member of the department of mathematics at Ben-Gurion University, Beer-Sheva, Israel. He is the incumbent of the Earl Katz Family chair in algebraic system theory. He has a double formation of electrical engineer (Telecom Paris, graduated 1978) and mathematician (PhD, Weizmann Institute, 1986). His research includes operator theory, stochastic analysis, and the theory of linear systems. Daniel Alpay is one of the initiators and responsible of the dual track electrical-engineering mathematics at Ben-Gurion University. Together with co-authors, he has written two books and close to 190 research papers, and edited ten books of research papers.

Policies and ethics