List of contributors ....................................... vii
Preface ..................................................... ix
1 John Williams Calkin: a short biography
S. Hassi, H.S.V. de Snoo and F.H. Szafraniec ................. 1
2 On Calkin's abstract symmetric boundary conditions
S. Hassi and H.L. Wietsma .................................... 3
2.1 Introduction ............................................ 3
2.2 Preliminaries ........................................... 5
2.3 Reduction operators .................................... 10
2.4 Maximal symmetric extensions and unbounded reduction
operators .............................................. 23
3 Boundary triplets and maximal accretive extensions of
sectorial operators
Y. Arlinskiĭ ................................................ 35
3.1 Introduction ........................................... 35
3.2 Preliminaries .......................................... 38
3.3 Friedrichs and Kreпn-von Neumann extensions ............ 43
3.4 Boundary pairs and closed forms associated with
m-sectorial extensions ................................. 45
3.5 Boundary triplets and m-accretive extensions ........... 47
3.6 WF- and QF-functions ................................... 52
3.7 Realization of the Phillips boundary space ............. 54
3.8 Vishik-Birman-Grubb type formulas ...................... 55
3.9 m-sectorial extensions via fractional-linear
transformations ........................................ 57
3.10 Sectorial operators in divergence form ................. 60
4 Boundary control state/signal systems and boundary
triplets
D.Z. Arov, M. Kurula and O.J. Staffans ...................... 73
4.1 Introduction ........................................... 73
4.2 Boundary control systems ............................... 74
4.3 Conservative state/signal systems in boundary control .. 76
4.4 An example: the transmission line ...................... 78
4.5 The connection to boundary triplets .................... 81
5 Passive state/signal systems and conservative boundary
relations
D.Z. Arov, M. Kurula and O.J. Staffans ...................... 87
5.1 Introduction ........................................... 87
5.2 Continuous-time state/signal systems ................... 88
5.3 Passive and conservative state/signal systems .......... 94
5.4 The frequency domain characteristics of a s/s system .. 104
5.5 Conservative boundary relations ....................... 107
5.6 Conclusions ........................................... 116
6 Elliptic operators, Dirichlet-to-Neumann maps and quasi
boundary triples
J. Behrndt and M. Langer ................................... 121
6.1 Introduction .......................................... 121
6.2 Boundary triples and Weyl functions for ordinary
and partial differential operators .................... 126
6.3 Quasi boundary triples and their Weyl functions ....... 135
6.4 Quasi boundary triples for elliptic operators and
Dirichlet-to-Neumann maps ............................. 147
7 Boundary triplets and Weyl functions. Recent developments
V.A. Derkach, S. Hassi, M.M. Malamud and H.S.V. de Snoo .... 161
7.1 Introduction .......................................... 161
7.2 Preliminaries ......................................... 165
7.3 Ordinary boundary triplets ............................ 172
7.4 Boundary triplets of bounded type ..................... 175
7.5 Boundary triplets of bounded type and infinite
dimensional graph perturbations ....................... 184
7.6 Unitary boundary relations and Weyl families .......... 188
7.7 Generalized resolvents and unitary boundary triplets .. 201
7.8 Isometric boundary mappings ........................... 205
8 Extension theory for elliptic partial differential
operators with pseudodifferential methods
G. Grubb ................................................... 221
8.1 Introduction .......................................... 221
8.2 Elliptic boundary value problems ...................... 222
8.3 Pseudodifferential operators .......................... 226
8.4 Pseudodifferential boundary operators ................. 228
8.5 Extension theories .................................... 232
8.6 Implementation of the abstract set-up for elliptic
operators ............................................. 237
8.7 Resolvent formulas .................................... 243
8.8 Applications of pseudodifferential methods I:
Conditions for lower boundedness ...................... 244
8.9 Applications of pseudodifferential methods II:
Spectral asymptotics .................................. 247
8.10 New spectral results .................................. 251
9 Dirac structures and boundary relations
S. Hassi, A.J. van der Schaft, H.S.V. de Snoo and
H.J. Zwart ................................................. 259
9.1 Introduction .......................................... 259
9.2 Linear relations in Hilbert and Kreĭn spaces .......... 259
9.3 Linear relations in product spaces .................... 263
9.4 The connections between various structures ............ 268
9.5 Weyl families and transfer functions .................. 271
10 Naĭmark dilations and Naĭmark extensions in favour
of moment problems
F.H. Szafraniec ............................................ 275
10.1 Dilations and extensions ............................. 276
10.2 The example .......................................... 287
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