This book presents the results obtained by the authors over the last four decades in the extension theory of symmetric operators in Hilbert spaces.

First, some classic results are highlighted, influenced primarily by Krein, Vishik and Birman. Then, the method of boundary triples is discussed, demonstrating the universal character of the Weyl function which arises naturally both in problems of mathematical physics and classical interpolation problems.

Readers of this book will gain an insight into the impressive construction of extension theory and its applications to problems in mathematical physics and analysis.

Provides a comprehensive review of the modern extension theory based on the concept of the boundary triple Presents a new approach to the theory of characterisic functions of unbounded operators Delivers a theory of functional models of non-self-adjoint extensions of a symmetric operator

Autorentext

Dr. Prof. Volodymyr Derkach started his research in Operator Theory in Donetsk University. His PhD thesis written in 1978 under the supervision of E. R. Tsekanovskii was devoted to characteristic functions for unbounded quasi-selfadjoint operators in indefinite inner product spaces. Professor Derkach has been working as a lecturer at Vasyl' Stus Donetsk National University (former Donetsk State University) more than 30 years. He was also working as a visiting professor at several universities in the USA and Germany. He was honored with Weston Visiting Scholarship grant (Weizmann Institute of Science, Israel), the Fulbright Research grant (University of Massachusetts Lowell, USA) and DFG Mercator professorship grant (Technical University of Ilmenau, Germany). V. Derkach is a member of the Editorial boards of Ukrainian Mathematical Bulletin and Carpathian Mathematical Publications.

Dr. Mark Malamud started his research in Operator Theory in Donetsk State University. His PhD thesis was written under supervision of Eduard Tsekanovskii and was defended in 1977. It was devoted to the similarity of Volterra operators. Dr. M. Malamud has been working as a lecturer at Donetsk Polytechnic Institute in 1978-1994 and in Donetsk National University in 1994-2007. Then he worked in the Institute of Applied Mathematics and Mechanics as a senior scientific investigator (2007-2011) and a leading scientific investigator (2011-2017). He was also working as a visiting professor at several universities in the USA, Germany, Italy, and France. M. Malamud is a member of the International Association of Mathematical Physics. Besides, he is a member of the Editorial boards of International journals: Mathematische Nachrichten and Methods of Functional Analysis and Topology. Collaboration of V. Derkach with M. Malamud started in the beginning of eighties. In a series of joint papers, they introduced the notion of the abstract Weyl function of a symmetric operator in a Hilbert space and applied it to spectral problems for differential operators, to Hamburger and Stieltjes moment problems, and to a large class of interpolation problems. These studies are summarized in their monograph "Extension Theory of Symmetric Operators and Boundary Value Problems" published in 2017 in the series of Proceedings of Institute of Mathematics of NAS of Ukraine (in Russian).

Inhalt

• 1. Introduction.- 2. Unbounded Linear Operators.- 3. Extension Theory of Symmetric Operators.- 4. Semibounded Operators and Forms.- 5. Block Operator Matrices and Kren's Extension Theory of Non-Negative Operators.- 6. HerglotzNevanlinna Functions.- 7. Linear Relations.- 8. Boundary Triples, Proper Extensions and Weyl Functions.- 9. Boundary Triples and Extension Theory of Non-Negative Operators.- 10. Boundary Triples and Extension Theory of Operators with Gaps.- 11. SturmLiouville Operator.- 12. Extension Theory for Non-Densely Defined Symmetric Operator.- 13. Functional Models of Symmetric and Selfadjoint Operators.- 14. Characteristic Functions and Resolvent Matrices.