Operator Methods for Boundary Value Problems

doi:10.1017/CBO9781139135061

Presented in this volume are a number of new results concerning the extension theory and spectral theory of unbounded operators using the recent notions of boundary triplets and boundary relations. This approach relies on linear single-valued and multi-valued maps, isometric in a Krein space sense, and offers a basic framework for recent developments in system theory. Central to the theory are analytic tools such as Weyl functions, including Titchmarsh-Weyl m-functions and Dirichlet-to-Neumann maps. A wide range of topics is considered in this context from the abstract to the applied, including boundary value problems for ordinary and partial differential equations; infinite-dimensional perturbations; local point-interactions; boundary and passive control state/signal systems; extension theory of accretive, sectorial and symmetric operators; and Calkin's abstract boundary conditions. This accessible treatment of recent developments, written by leading researchers, will appeal to a broad range of researchers, students and professionals.

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Contents

Frontmatter
pp i-iv
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Contents
pp v-viii
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List of contributors
pp ix-x
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Preface
pp xi-xii
- By Seppo Hassi, University of Vaasa, Henk de Snoo, University of Groningen, Franek Szafraniec, Uniwersytet Jagielloński
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1 - John Williams Calkin: a short biography
pp 1-2
- By S. Hassi, University of Vaasa, H.S.V. de Snoo, University of Groningen, F.H. Szafraniec, Uniwersytet Jagielloński
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2 - On Calkin's abstract symmetric boundary conditions
pp 3-34
- By S. Hassi, University of Vaasa, H.L. Wietsma, University of Vaasa
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3 - Boundary triplets and maximal accretive extensions of sectorial operators
pp 35-72
- By Y. Arlinskiĭ, East Ukrainian National University
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4 - Boundary control state/signal systems and boundary triplets
pp 73-86
- By D.Z. Arov, Institute of Physics and Mathematics, M. Kurula, University of Twente, O.J. Staffans, Åbo Akademi University
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5 - Passive state/signal systems and conservative boundary relations
pp 87-120
- By D.Z. Arov, South-Ukrainian National Pedagogical University, M. Kurula, University of Twente, O.J. Staffans, Åbo Akademi University,
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6 - Elliptic operators, Dirichlet-to-Neumann maps and quasi boundary triples
pp 121-160
- By J. Behrndt, Technische Universität Graz, M. Langer, University of Strathclyde
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7 - Boundary triplets and Weyl functions. Recent developments
pp 161-220
- By V.A. Derkach, Donetsk National University, S. Hassi, University of Vaasa, M.M. Malamud, Donetsk National University, H.S.V. de Snoo, University of Groningen
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8 - Extension theory for elliptic partial differential operators with pseudodifferential methods
pp 221-258
- By G. Grubb, University of Copenhagen
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9 - Dirac structures and boundary relations
pp 259-274
- By S. Hassi, University of Vaasa, A.J. van der Schaft, University of Groningen, H.S.V. de Snoo, University of Groningen, H.J. Zwart, University of Twente
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10 - Naĭmark dilations and Naĭmark extensions in favour of moment problems
pp 275-298
- By F.H. Szafraniec, Uniwersytet Jagielloński
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Operator Methods for Boundary Value Problems

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Contents

Frontmatter
pp i-iv

Contents
pp v-viii

List of contributors
pp ix-x

Preface
pp xi-xii

1 - John Williams Calkin: a short biography
pp 1-2

2 - On Calkin's abstract symmetric boundary conditions
pp 3-34

3 - Boundary triplets and maximal accretive extensions of sectorial operators
pp 35-72

4 - Boundary control state/signal systems and boundary triplets
pp 73-86

5 - Passive state/signal systems and conservative boundary relations
pp 87-120

6 - Elliptic operators, Dirichlet-to-Neumann maps and quasi boundary triples
pp 121-160

7 - Boundary triplets and Weyl functions. Recent developments
pp 161-220

8 - Extension theory for elliptic partial differential operators with pseudodifferential methods
pp 221-258

9 - Dirac structures and boundary relations
pp 259-274

10 - Naĭmark dilations and Naĭmark extensions in favour of moment problems
pp 275-298

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Operator Methods for Boundary Value Problems

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