Boundary value problems, Weyl functions, and differential operators

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(1)Monographs in Mathematics 108. Jussi Behrndt Seppo Hassi Henk de Snoo. Boundary Value Problems, Weyl Functions, and Differential Operators.

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(3) Monographs in Mathematics Volume 108. Series Editors Herbert Amann, Universität Zürich, Zürich, Switzerland Jean-Pierre Bourguignon, IHES, Bures-sur-Yvette, France William Y. C. Chen, Nankai University, Tianjin, China Associate Editor Huzihiro Araki, Kyoto University, Kyoto, Japan John Ball, Heriot-Watt University, Edinburgh, UK Franco Brezzi, Università degli Studi di Pavia, Pavia, Italy Kung Ching Chang, Peking University, Beijing, China Nigel Hitchin, University of Oxford, Oxford, UK Helmut Hofer, Courant Institute of Mathematical Sciences, New York, USA Horst Knörrer, ETH Zürich, Zürich, Switzerland Don Zagier, Max-Planck-Institut, Bonn, Germany. The foundations of this outstanding book series were laid in 1944. Until the end of the 1970s, a total of 77 volumes appeared, including works of such distinguished mathematicians as Carathéodory, Nevanlinna and Shafarevich, to name a few. The series came to its name and present appearance in the 1980s. In keeping its well-established tradition, only monographs of excellent quality are published in this collection. Comprehensive, in-depth treatments of areas of current interest are presented to a readership ranging from graduate students to professional mathematicians. Concrete examples and applications both within and beyond the immediate domain of mathematics illustrate the import and consequences of the theory under discussion.. More information about this series at http://www.springer.com/series/4843.

(4) Jussi Behrndt Seppo Hassi Henk de Snoo •. •. Boundary Value Problems, Weyl Functions, and Differential Operators.

(5) Jussi Behrndt Institut für Angewandte Mathematik Technische Universität Graz Graz, Austria. Seppo Hassi Mathematics and Statistics University of Vaasa Vaasa, Finland. Henk de Snoo Bernoulli Institute for Mathematics Computer Science and Artificial Intelligence University of Groningen Groningen, The Netherlands. ISSN 1017-0480 ISSN 2296-4886 (electronic) Monographs in Mathematics ISBN 978-3-030-36713-8 ISBN 978-3-030-36714-5 (eBook) https://doi.org/10.1007/978-3-030-36714-5 Mathematics Subject Classification (2010): 47A, 47B, 47E, 47F, 34B, 34L, 35P, 81C, 93B © The Editor(s) (if applicable) and The Author(s) 2020. This book is an open access publication. Open Access This book is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made. The images or other third party material in this book are included in the book's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the book's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This book is published under the imprint Birkhäuser, www.birkhauser-science.com by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland.

(6) Contents. Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. ix. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 1. 1 Linear Relations in Hilbert Spaces 1.1 Elementary facts about linear relations . . . . . . . . . 1.2 Spectra, resolvent sets, and points of regular type . . . 1.3 Adjoint relations . . . . . . . . . . . . . . . . . . . . . . 1.4 Symmetric relations . . . . . . . . . . . . . . . . . . . . 1.5 Self-adjoint relations . . . . . . . . . . . . . . . . . . . . 1.6 Maximal dissipative and accumulative relations . . . . . 1.7 Intermediate extensions and von Neumann’s formulas . 1.8 Adjoint relations and indefinite inner products . . . . . 1.9 Convergence of sequences of relations . . . . . . . . . . 1.10 Parametric representations for relations . . . . . . . . . 1.11 Resolvent operators with respect to a bounded operator 1.12 Nevanlinna families and their representations . . . . . . 2. 3. Boundary Triplets and Weyl Functions 2.1 Boundary triplets . . . . . . . . . . . . . . . . . 2.2 Boundary value problems . . . . . . . . . . . . . 2.3 Associated γ-fields and Weyl functions . . . . . . 2.4 Existence and construction of boundary triplets 2.5 Transformations of boundary triplets . . . . . . 2.6 Kre˘ın’s formula for intermediate extensions . . . 2.7 Kre˘ın’s formula for exit space extensions . . . . 2.8 Perturbation problems . . . . . . . . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . .. . 11 . 23 . 30 . 42 . 48 . 58 . 65 . 74 . 79 . 87 . 96 . 100. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. 107 115 118 126 134 148 155 163. Spectra, Simple Operators, and Weyl Functions 3.1 Analytic descriptions of minimal supports of Borel measures . . . 169 3.2 Growth points of finite Borel measures . . . . . . . . . . . . . . . . 178 3.3 Spectra of self-adjoint relations . . . . . . . . . . . . . . . . . . . . 183.

(7) vi. Contents. 3.4 3.5 3.6 3.7 3.8 4. 5. 6. 7. Simple symmetric operators . . . . . . . . . . . . . . . . Eigenvalues and eigenspaces . . . . . . . . . . . . . . . . Spectra and local minimality . . . . . . . . . . . . . . . Limit properties of Weyl functions . . . . . . . . . . . . Spectra and local minimality for self-adjoint extensions. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. 188 196 203 212 218. Operator Models for Nevanlinna Functions 4.1 Reproducing kernel Hilbert spaces . . . . . . . . . . . . . . . . 4.2 Realization of uniformly strict Nevanlinna functions . . . . . . 4.3 Realization of scalar Nevanlinna functions via L2 -space models 4.4 Realization of Nevanlinna pairs and generalized resolvents . . . 4.5 Kre˘ın’s formula for exit space extensions . . . . . . . . . . . . 4.6 Orthogonal coupling of boundary triplets . . . . . . . . . . . .. . . . . . .. . . . . . .. 223 235 252 261 270 274. Boundary Triplets and Boundary Pairs for Semibounded Relations 5.1 Closed semibounded forms and their representations . . . . . 5.2 Ordering and monotonicity . . . . . . . . . . . . . . . . . . . 5.3 Friedrichs extensions of semibounded relations . . . . . . . . 5.4 Semibounded self-adjoint extensions and their lower bounds . 5.5 Boundary triplets for semibounded relations . . . . . . . . . 5.6 Boundary pairs and boundary triplets . . . . . . . . . . . . .. . . . . . .. . . . . . .. 282 300 311 319 332 343. . . . . . . . . .. . . . . . . . . .. 366 380 388 397 412 421 425 434 442. . . . . . .. Sturm–Liouville Operators 6.1 Sturm–Liouville differential expressions . . . . . . . . . . . . . 6.2 Maximal and minimal Sturm–Liouville differential operators . 6.3 Regular and limit-circle endpoints . . . . . . . . . . . . . . . . 6.4 The case of one limit-point endpoint . . . . . . . . . . . . . . . 6.5 The case of two limit-point endpoints and interface conditions 6.6 Exit space extensions . . . . . . . . . . . . . . . . . . . . . . . 6.7 Weyl functions and subordinate solutions . . . . . . . . . . . . 6.8 Semibounded Sturm–Liouville expressions in the regular case . 6.9 Closed semibounded forms for Sturm–Liouville equations . . . 6.10 Principal and nonprincipal solutions of Sturm–Liouville equations . . . . . . . . . . . . . . . . . . . 6.11 Semibounded Sturm–Liouville operators and the limit-circle case . . . . . . . . . . . . . . . . . . . . . . . . 6.12 Semibounded Sturm–Liouville operators and the limit-point case . . . . . . . . . . . . . . . . . . . . . . . . 6.13 Integrable potentials . . . . . . . . . . . . . . . . . . . . . . . .. . . 454 . . 469 . . 477 . . 483. Canonical Systems of Differential Equations 7.1 Classes of integrable functions . . . . . . . . . . . . . . . . . . . . 500 7.2 Canonical systems of differential equations . . . . . . . . . . . . . 504.

(8) Contents. 7.3 7.4 7.5 7.6 7.7 7.8 7.9 7.10 8. vii. Regular and quasiregular endpoints . . . . . . . . . . . . Square-integrability of solutions of real canonical systems Definite canonical systems . . . . . . . . . . . . . . . . . . Maximal and minimal relations for canonical systems . . Boundary triplets for the limit-circle case . . . . . . . . . Boundary triplets for the limit-point case . . . . . . . . . Weyl functions and subordinate solutions . . . . . . . . . Special classes of canonical systems . . . . . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. 510 513 520 525 534 543 559 566. Schr¨ odinger Operators on Bounded Domains 8.1 Rigged Hilbert spaces . . . . . . . . . . . . . . . . . . . . 8.2 Sobolev spaces, C 2 -domains, and trace operators . . . . . 8.3 Trace maps for the maximal Schr¨ odinger operator . . . . 8.4 A boundary triplet for the maximal Schrödinger operator 8.5 Semibounded Schr¨ odinger operators . . . . . . . . . . . . 8.6 Coupling of Schr¨ odinger operators . . . . . . . . . . . . . 8.7 Bounded Lipschitz domains . . . . . . . . . . . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. 577 581 588 600 611 616 624. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. 631 636 645 655 663 668. A Integral Representations of Nevanlinna Functions A.1 Borel transforms and their Stieltjes inversion A.2 Scalar Nevanlinna functions . . . . . . . . . . A.3 Operator-valued integrals . . . . . . . . . . . A.4 Operator-valued Nevanlinna functions . . . . A.5 Kac functions . . . . . . . . . . . . . . . . . . A.6 Stieltjes and inverse Stieltjes functions . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. . . . . . .. B Self-adjoint Operators and Fourier Transforms B.1 The scalar case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 678 B.2 The vector case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 685 C Sums of Closed Subspaces in Hilbert Spaces . . . . . . . . . . . . . . . . 691 D Factorization of Bounded Linear Operators . . . . . . . . . . . . . . . . 699 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 705 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 721 List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 765 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 769.

(9) Preface This monograph is about boundary value problems, Weyl functions, and differential operators. It grew out of a number of courses and seminars on functional analysis, operator theory, and differential equations, which the authors have given over a long period of time at various institutions. The project goes back to 2005 with a course on extension theory of symmetric operators, boundary triplets, and Weyl functions given at TU Berlin, while an extended form of the course was presented in 2006/2007 at the University of Groningen. Many more such courses and seminars, often on special topics, would follow at TU Berlin, Jagiellonian University in Krak´ ow, and, since 2011, at TU Graz. The authors wish to thank all the students, PhD students, and postdocs who have attended these lectures; their critical questions and comments have led to numerous improvements. They have shown that lectures at the blackboard provide the ultimate test for the quality of the material. In particular, we mention Bernhard Gsell, Markus Holzmann, Christian K¨ uhn, Vladimir Lotoreichik, Jonathan Rohleder, Peter Schlosser, Philipp Schmitz, Simon Stadler, Alef Sterk, and Rudi Wietsma. It is our experience that the individual chapters of this monograph can be used (with small additions from some of the other chapters) for independent courses on the respective topics. The book has benefited from our collaboration with many different colleagues. We would like to single out our friends and faithful coauthors Yuri Arlinski˘ı, Vladimir Derkach, Peter Jonas, Matthias Langer, Annemarie Luger, Mark Malamud, Hagen Neidhardt, Franek Szafraniec, Carsten Trunk, Henrik Winkler, and Harald Woracek. Special thanks go to Fritz Gesztesy, Gerd Grubb, Heinz Langer, and James Rovnyak, who have responded to our queries concerning historical developments and references. We gratefully acknowledge the support of the following institutions: Deutsche Forschungsgemeinschaft, Jagiellonian University, TU Berlin, and TU Graz. We would like to thank the Mathematisches Forschungsinstitut Oberwolfach and the Mittag-Leffler Institute in Djursholm for their hospitality during the final stages of the preparation of this book. Finally, we are indebted to the Austrian Science Fund (Grant PUB 683-Z) and the University of Vaasa for funding the open access publication of this monograph. Jussi Behrndt, Seppo Hassi, and Henk de Snoo.

(10) Introduction In this monograph the theory of boundary triplets and their Weyl functions is developed and applied to the analysis of boundary value problems for differential equations and general operators in Hilbert spaces. Concrete illustrations by means of weighted Sturm–Liouville differential operators, canonical systems of differential equations, and multidimensional Schr¨ odinger operators are provided. The abstract notions of boundary triplets and Weyl functions have their roots in the theory of ordinary differential operators; they appear in a slightly different context also in the treatment of partial differential operators. Before describing the contents of the monograph it may be helpful to explain the ideas in this text by means of the following simple Sturm–Liouville differential expression d2 L = − 2 + V, (1) dx where it is assumed that the potential V is a real measurable function. The context in which this differential expression will be placed serves as an example as well as a motivation. The first step is to associate with L some differential operators in a suitable Hilbert space. Assume, e.g., that (1) is given on the positive half-line R+ = (0, ∞) and assume for simplicity that the real function V is bounded. Define the linear space Dmax by Dmax = f ∈ L2 (R+ ) : f, f absolutely continuous, Lf ∈ L2 (R+ ) and define the minimal operator S associated with L by dom S = f ∈ Dmax : f (0) = f (0) = 0 . Sf = −f + V f, Then S is a closed densely defined symmetric operator L2 (R+ ); in fact, it is the closure of (the graph of) the restriction of S to C0∞ (R+ ). It can be shown that the adjoint operator S ∗ is given by S ∗ f = −f + V f,. dom S ∗ = Dmax ,. which is usually called the maximal operator associated with L. Roughly speaking, S is a two-dimensional restriction of S ∗ by means of the boundary conditions. © The Author(s) 2020 J. Behrndt et al., Boundary Value Problems, Weyl Functions, and Differential Operators, Monographs in Mathematics 108, https://doi.org/10.1007/978-3-030-36714-5_1. 1.

(11) 2. Introduction. f (0) = 0 and f (0) = 0. Note that the maximal domain Dmax coincides with the second-order Sobolev space H 2 (R+ ). The notion of boundary triplet will now be explained in the present situation. For this consider f, g ∈ dom S ∗ and observe that integration by parts leads to ∞ ∞ (S ∗ f, g)L2 (R+ ) − (f, S ∗ g)L2 (R+ ) = −f (x)g(x) +f (x)g (x) 0. 0. = f (0)g(0) − f (0)g (0), where it was used that the products f g and f g vanish at ∞. Inspired by the above identity, define boundary mappings Γ0 , Γ1 : dom S ∗ → C,. f → Γ0 f := f (0). and. f → Γ1 f := f (0),. (2). so that for all f, g ∈ dom S ∗ one has (S ∗ f, g)L2 (R+ ) − (f, S ∗ g)L2 (R+ ) = (Γ1 f, Γ0 g)C − (Γ0 f, Γ1 g)C ,. (3). which is the so-called abstract Green identity in the definition of a boundary triplet; note that on the right-hand side of (3) the scalar product in the (boundary) Hilbert space C is used. This abstract Green identity is the key feature in the notion of a boundary triplet and it is primarily responsible for the succesful functioning of the whole theory. Note also that the combined boundary mapping (Γ0 , Γ1 ) : dom S ∗ → C2 is surjective, which is understood as a maximality condition in the sense that the image space of the boundary maps is not unnecessarily large. Observe that one has dom S = ker Γ0 ∩ ker Γ1 . The operator realizations A of the Sturm–Liouville differential expression L which are intermediate extensions, that is, S ⊂ A ⊂ S ∗ , can be described by boundary conditions expressed via the boundary maps. More precisely, for τ ∈ C ∪ {∞} the operator Aτ is defined by Aτ f = S ∗ f,. dom Aτ = ker (Γ1 − τ Γ0 ),. (4). which in a more explicit form reads Aτ f = −f + V f,. dom Aτ = f ∈ Dmax : f (0) = τ f (0) ;. the case τ = ∞ is understood as the boundary condition ker Γ0 , that is, dom A∞ = f ∈ Dmax : f (0) = 0 . A∞ f = −f + V f,. (5). In the definition (4) the quantity τ plays the role of a boundary parameter that links the boundary values Γ0 f = f (0) and Γ1 f = f (0) of the functions f ∈ dom S ∗ , which determine the Dirichlet and Neumann boundary conditions, respectively. The properties of the boundary parameter are directly connected with.

(12) Introduction. 3. the properties of the corresponding operator Aτ ; in particular, the realization Aτ is self-adjoint in L2 (R+ ) if and only if τ ∈ R ∪ {∞}. The next main goal is to motivate and illustrate the definition of the Weyl function as an analytic object corresponding to a boundary triplet, which is indispensable in the spectral theory of the intermediate extensions. For this, let λ ∈ C and consider first the unique solutions ϕλ and ψλ of the boundary value problems −ϕλ + V ϕλ = λϕλ , −ψλ + V ψλ = λψλ ,. ϕλ (0) = 1, ψλ (0) = 0,. ϕλ (0) = 0, ψλ (0) = 1,. (6). and note that in general ϕλ , ψλ ∈ L2 (R+ ). It was shown by H. Weyl more than a century ago that for λ ∈ C \ R there exists m(λ) ∈ C such that x → fλ (x) = ϕλ (x) + m(λ)ψλ (x) ∈ L2 (R+ ),. (7). and it turned out that the function m : C \ R → C is holomorphic and has a positive imaginary part in the upper half-plane C+ . This function and its interplay with spectral theory were later studied extensively by E.C. Titchmarsh; hence the frequently used terminology Titchmarsh–Weyl m-function. It plays a key role in the spectral analysis of Sturm–Liouville differential operators. E.g., the (real) poles of m coincide with the isolated eigenvalues of the self-adjoint Dirichlet operator A∞ in (5) and the absolutely continuous spectrum of A∞ is, roughly speaking, given by those λ ∈ R for which Im m(λ + i0) > 0. In a similar way one can also characterize the continuous spectrum, the embedded eigenvalues, and exclude singular continuous spectrum of A∞ . Observe that for each λ ∈ C \ R the function x → fλ (x) in (7) belongs to dom S ∗ = Dmax and that, in fact, −fλ + V fλ = λfλ for λ ∈ C \ R; in other words, fλ ∈ ker (S ∗ − λ). Let {C, Γ0 , Γ1 } be the boundary triplet for S ∗ with the boundary mappings defined in (2). From the choice of ϕλ and ψλ in (6) it is clear that (8) m(λ)Γ0 fλ = m(λ)fλ (0) = m(λ) = Γ1 fλ , fλ ∈ ker (S ∗ − λ). In the general theory this identity is used as the definition of the Weyl function corresponding to a boundary triplet. In other words, the Weyl function corresponding to the boundary triplet {C, Γ0 , Γ1 } is defined as the function m that satisfies (8) for all λ ∈ C \ R (and even for the possibly larger set of λ belonging to the resolvent set of the self-adjoint Dirichlet operator A∞ ) and hence coincides with the Titchmarsh–Weyl m-function introduced via (7). Here the Weyl function maps Dirichlet boundary values of L2 -solutions of the equation −fλ + V fλ = λfλ onto the corresponding Neumann boundary values and therefore m(λ) acts formally like a Dirichlet-to-Neumann map. Besides the Weyl function, one associates to the boundary triplet {C, Γ0 , Γ1 } the so-called γ-field as the mapping γ(λ) : C → L2 (R+ ) that assigns to a prescribed boundary value c ∈ C the solution hλ ∈ dom S ∗ of the boundary value problem −hλ + V hλ = λhλ ,. Γ0 hλ = hλ (0) = c..

(13) 4. Introduction. Since γ(λ)c = hλ = cfλ , it is clear that m(λ) = Γ1 γ(λ). Moreover, one can show with the help of the abstract Green identity that the adjoint γ(λ)∗ : L2 (R+ ) → C is given by γ(λ)∗ = Γ1 (A∞ −λ)−1 . The Weyl function and γ-field associated to the boundary triplet {C, Γ0 , Γ1 } appear in the perturbation term in Kre˘ın’s formula (Aτ − λ)−1 = (A∞ − λ)−1 + γ(λ)(τ − m(λ))−1 γ(λ)∗ , where, for simplicity, it is assumed that Aτ is a self-adjoint realization of L as in (4) corresponding to some boundary parameter τ ∈ R and λ ∈ ρ(Aτ )∩ρ(A∞ ). Kre˘ın’s formula in this particular case provides a description of the resolvent difference of Aτ and the fixed self-adjoint extension A∞ . It is important to note that γ(λ) and γ(λ)∗ in the perturbation term provide a link between the original Hilbert space L2 (R+ ) and the boundary space C, but do not affect the resolvents of A∞ and Aτ . Therefore, if λ ∈ ρ(A0 ), then the singularities of the resolvent λ → (Aτ − λ)−1 are reflected in the singularities of the term λ → (τ − m(λ))−1 and vice versa. In fact, the function λ → (τ − m(λ))−1 is connected with the spectrum of Aτ in the same way as the function λ → m(λ) is connected with the spectrum of A∞ . There is another efficient technique to associate differential operators with the differential expression L, which is based on the sesquilinear form t corresponding to L, t[f, g] = (f , g )L2 (R+ ) + (V f, g)L2 (R+ ) , (9) defined on, e.g., D = f ∈ L2 (R+ ) : f absolutely continuous, f ∈ L2 (R+ ) ,. (10). and the first representation theorem for sesquilinear forms. In fact, one verifies that t in (9)–(10) is a densely defined closed semibounded form in L2 (R+ ), and hence there exists a uniquely determined self-adjoint operator S1 with dom S1 ⊂ D such that f ∈ dom S1 , g ∈ dom D. (11) (S1 f, g)L2 (R+ ) = t[f, g], Note that here the form domain D coincides with the first-order Sobolev space H 1 (R+ ). It can be shown that the self-adjoint operator S1 is actually an extension of the minimal operator S. Instead of the domain D in (10) one may consider the sesquilinear form t on the smaller domain D0 = {f ∈ D : f (0) = 0}, which also leads to a densely defined closed semibounded form in L2 (R+ ). Again, via the first representation theorem, there is a corresponding self-adjoint operator S0 with dom S0 ⊂ D0 determined by (S0 f, g)L2 (R+ ) = t[f, g],. f ∈ dom S0 , g ∈ dom D0 .. (12). One verifies that the self-adjoint operator S1 in (11) coincides with the self-adjoint realization of L determined by the boundary condition ker Γ1 and that the selfadjoint operator S0 in (12) coincides with the self-adjoint realization of L determined by the boundary condition ker Γ0 in (4), that is, S1 corresponds to the.

(14) Introduction. 5. boundary parameter τ = 0 and S0 is the Dirichlet operator corresponding to the boundary parameter τ = ∞. Furthermore, in the situation discussed here the self-adjoint operator S0 in (12) is the Friedrichs extension of the minimal (or preminimal) operator associated to L. The concept of boundary triplet is supplemented by the notion of boundary pair, which is inspired by the form approach indicated above. More precisely, in the present situation it turns out that {G, Λ}, where G = C and Λ : D → C,. f → Λf := f (0),. (13). is a boundary pair for the minimal operator S (corresponding to S1 ). For this, one has to ensure that the mapping Λ defined on the form domain of S1 is continuous with respect to the Hilbert space topology generated by the closed form t on D, and that ker Λ coincides with the form domain corresponding to the Friedrichs extension of S. Note also that in the present situation the mapping Λ in (13) is an extension of the boundary mapping Γ0 : dom S ∗ → C to the form domain D. With the help of the boundary pair {C, Λ} one can parametrize all densely defined closed semibounded forms corresponding to semibounded self-adjoint extensions of S via tτ [f, g] = t[f, g] + (τ Λf, Λg)C ,. f, g ∈ D,. (14). where τ ∈ R ∪ {∞}, and the case τ = ∞ corresponds to the boundary condition Λf = 0 in D0 . The boundary pair and the boundary triplet are connected via the first Green identity (S ∗ f, g)L2 (R+ ) = t[f, g] + (Γ1 f, Λg)C ,. f ∈ dom S ∗ , g ∈ D.. The first Green identity makes it possible to identify the closed semibounded forms in (14) with the corresponding self-adjoint operator realizations Aτ of L described via boundary conditions in (4). For f ∈ dom Aτ and g ∈ D, the first Green identity reduces to (Aτ f, g)L2 (R+ ) = t[f, g] + (τ Γ0 f, Λg)C = t[f, g] + (τ Λf, Λg)C , and the expression (τ Λf, Λg)C on the right-hand side can also be interpreted as a sesquilinear form in the boundary space C. In this sense the theory of boundary pairs for semibounded symmetric operators complements the theory of boundary triplets in a natural way: it provides a description of the closed semibounded forms corresponding to semibounded self-adjoint extensions of the minimal operator S. Methods to treat Sturm–Liouville problems such as the one discussed above go back to H. Weyl [758, 759, 760], whose papers on this topic appeared in 1910/1911; see also [761]. The interpretation of a Sturm–Liouville expression as an operator in a Hilbert space can already be found in the 1932 book of M.H. Stone [724]. In this monograph Stone gave an abstract treatment of operators in a Hilbert space including the work of J. von Neumann [610, 611] from 1929 and 1932, who.

(15) 6. Introduction. had also introduced the extension theory of densely defined symmetric operators and found the formulas which carry his name: self-adjoint extensions correspond to unitary mappings between the defect spaces. The von Neumann formulas are abstract, since they are formulated in terms of the defect spaces of the symmetric operator, and they needed to be related to concrete boundary value problems. With this in mind another approach involving abstract boundary conditions was developed by J.W. Calkin [187] in his 1937 Harvard doctoral dissertation, which was written under the direction of Stone, who suggested the topic. Calkin was also advised by von Neumann. Calkin’s work on boundary value problems did not receive the attention it might have deserved. It seems that he never returned to it; his later mathematical work was related to World War II and the Manhattan project in Los Alamos. Another way to deal with the self-adjoint extensions of a symmetric operator is via Kre˘ın’s resolvent formula. The early background of this formula can be found in the idea of perturbation of self-adjoint operators. Kre˘ın’s formula describes the resolvent of a self-adjoint extension in terms of the resolvent of a fixed self-adjoint extension and a perturbation term which involves a so-called Q-function and a parameter describing the self-adjoint extension. The Q-function uniquely determines the underlying symmetry and the fixed self-adjoint extension, up to unitary equivalence, and thus reflects their spectral properties. The original Kre˘ın formula for equal finite defect numbers goes back to M.G. Kre˘ın [491, 492] in the middle of the 1940s; only in 1965 it was finally established for the case of equal infinite defect numbers by S.N. Saakyan [679]. In fact, the self-adjoint extensions were allowed to be in a Hilbert space which contains the original Hilbert space as a closed subspace. This type of extension appeared after 1940 in papers by M.G. Kre˘ın and ˇ M.A. Na˘ımark [605, 606, 607]. Later A.V. Straus in the 1950s and 1960s described such exit space extensions in the framework of the von Neumann formulas via holomorphic contractions between the defect spaces [731]. The Q-function in Kre˘ın’s formula can be seen as an abstract analog of the Titchmarsh–Weyl function in the above Sturm–Liouville example; it was extensively studied in the 1960s and 1970s by M.G. Kre˘ın and H. Langer [497]–[504], also in the context of Pontryagin spaces. From the early 1940s on E.C. Titchmarsh turned his attention to the singular Sturm–Liouville equation. He put aside Weyl’s method of handling the Sturm– Liouville problem on the basis of integral equations and also bypassed the use of the general theory of linear operators in Hilbert spaces as in Stone’s book [739]. Instead, Titchmarsh used contour integration and the Cauchy calculus of residues, influenced by the work of E. Hilb [417, 418, 419], a contemporary of Weyl. In this way he found a simple formula to determine the spectral measure; this last formula was also discovered by K. Kodaira around the same time [469, 470]. A complete survey of the work of Titchmarsh, both for ordinary and partial differential operators, is given in his two books on eigenfunction expansions [740, 741]. A different approach, followed by B.M. Levitan [541, 542], N. Levinson [539, 540], and K. Yosida [780, 781], is based on the fact that the resolvent operator of the.

(16) Introduction. 7. self-adjoint realization of a singular differential operator can be approximated by compact resolvents corresponding to Sturm–Liouville problems for proper closed subintervals. Closely connected with this is an abstract approach to eigenfunction expansions generated by differential operators that was introduced by Kre˘ın [495] in the form of directing functionals. Influenced by questions from mathematical physics, von Neumann posed the following problem in the middle of the 1930s: can one extend a densely defined semibounded symmetric operator to a self-adjoint operator with the same lower bound? There were contributions by M.H. Stone [724] and K.O. Friedrichs [310] (whose work was simplified by H. Freudenthal [309]). The Friedrichs extension was the solution to von Neumann’s problem. For Sturm–Liouville operators the Friedrichs extension was determined in various cases by K.O. Friedrichs [311] in 1935 and by F. Rellich [654] in 1950. Another semibounded extension, the so-called Kre˘ın–von Neumann extension (going back to Stone) has particularly interesting properties. It was Kre˘ın [493, 494] who established a complete theory of semibounded extensions. In the middle of the 1950s this circle of ideas was carried forward, and it inspired contributions by M.S. Birman [139], and also M.I. Vishik [747], who was particularly interested in the case of elliptic partial differential operators. Building on the work of J.L. Lions and E. Magenes [544] on Sobolev spaces and trace mappings G. Grubb [352, 353] gave a characterization of all closed extensions of a minimal elliptic operator by nonlocal boundary conditions in her 1966 Stanford doctoral dissertation, written under the direction of R.S. Phillips. The context of symmetric operators which are densely defined was soon felt to be too restrictive. Already in 1949 M.A. Krasnoselski˘ı [490] described all selfadjoint operator extensions of a not necessarily densely defined symmetric operator. The appearance of the work on linear relations by R. Arens [42] in 1961 made all the difference. B.C. Orcutt [619] in a 1969 dissertation written under the direction of J. Rovnyak treated the spectral theory of canonical systems of differential equations in terms of linear relations. Subsequently, E.A. Coddington [202] in 1973 gave a description of all self-adjoint relation extensions of a symmetric relation. In fact, it turned out that many of the earlier results concerning extensions of symmetric operators could be put in the framework of relations. The new context made it also possible to consider nonstandard boundary conditions (involving integrals, for instance). Furthermore, in terms of relations the Kre˘ın– von Neumann extension of a semibounded relation could be simply expressed in terms of the Friedrichs extension. There has been an abundance of papers devoted to linear relations in Hilbert spaces, and later also to linear relations in indefinite inner product spaces. In the middle of the 1970s boundary triplets were introduced independently by V.M. Bruk [176] and A.N. Kochubei [466] as a convenient tool for the description of boundary values of abstract Hilbert space operators; they applied them to, e.g., Sturm–Liouville operators with an operator-valued potential. The main feature is that under a given boundary triplet there is a natural correspondence.

(17) 8. Introduction. between self-adjoint extensions of a symmetric operator and self-adjoint relations in the parameter space. An overview of the theory with applications to differential operators is contained in the 1984 book by M.L. Gorbachuk and V.I. Gorbachuk [346]. Around the same time V.A. Derkach and M.M. Malamud [244, 246] continued the work on boundary triplets by associating the notion of Weyl function to a boundary triplet; their later work was written in the context of symmetric operators that are not necessarily densely defined. The Weyl function is a very useful tool in spectral analysis; it turns out to be a special choice of a Q-function (which is uniquely determined by the boundary triplet) and hence the analytic properties and the limit behavior of the Weyl function towards the real line reflect the spectral properties of the self-adjoint extensions. Broadly speaking, boundary triplets and Weyl functions placed the work of Titchmarsh, and others, in a more abstract setting while retaining the flavor of concrete boundary value problems. The link to form methods and the Birman–Kre˘ın–Vishik approach to semibounded self-adjoint extensions is made with the help of so-called boundary pairs. The origin of the concept of boundary pair lies in the work of Kre˘ın and Vishik; it was formalized and studied by V.E. Lyantse and O.G. Storozh [552] in the early 1980s. Its connection with boundary triplets was later established by Yu.M. Arlinski˘ı [44]. It is the main objective of this monograph to present the theory of boundary triplets and Weyl functions in an easily accessible and self-contained manner. The exposition is detailed and kept as simple as possible; the reader is only assumed to be familiar with the basic principles of functional analysis and some fundamentals of the spectral theory of self-adjoint operators in Hilbert spaces. The monograph is divided into the abstract part Chapters 1–5, the applied part Chapters 6–8, and Appendices A–D. The heart of the monograph is Chapter 2 and it is complemented by Chapter 5; for a rough idea on the general techniques the reader may first look through these chapters and examine one of the applications (which may also be read independently) afterwards: Sturm–Liouville operators, canonical systems, or Schr¨ odinger operators – up to personal taste and preferences.. The monograph opens in Chapter 1 with a detailed introduction to the theory of linear operators and relations in Hilbert spaces. A large part of this material is preparatory and may be used for reference purposes in the rest of the text..

(18) Introduction. 9. The heart of the matter in this book is contained in Chapter 2, where boundary value problems are presented as extension problems of symmetric operators or relations. Here the notions of boundary triplets and their Weyl functions are introduced, and the fundamental properties of these objects are provided. Particular attention is paid to the question of existence and uniqueness of boundary triplets. Closely connected with a boundary triplet is Kre˘ın’s resolvent formula for canonical extensions and self-adjoint extensions in larger Hilbert spaces. Chapter 3 is a continuation and further refinement of the techniques in the previous chapter. Here the main objective is to give a detailed description of the complete spectrum of the self-adjoint extensions of a symmetric relation in terms of the Weyl function. The connection between the limit properties of the Weyl function and the spectrum of the self-adjoint extension is explained via the Borel transform of the spectral measure. Most of the topics in Chapter 4 are supplementary to the main text as they are concerned with a certain type of inverse problem. More precisely, it will be shown that any (uniformly strict) operator-valued Nevanlinna function can be realized as the Weyl function corresponding to a boundary triplet for a symmetric relation in a reproducing kernel Hilbert model space. Of independent interest is the discussion around the orthogonal coupling of boundary triplets with a view to exit space extensions. Another central theme in this monograph is presented in Chapter 5, where the important case of semibounded symmetric relations is treated in more detail; here the general methods from Chapter 2 are further developed. The chapter starts with an introduction to closed semibounded forms and the corresponding representation theorems, and continues with the Friedrichs extension, the so-called Kre˘ın type extensions, and the Kre˘ın–von Neumann extension. The ultimate result is a description of the semibounded self-adjoint extensions of a semibounded relation via the notions of a boundary triplet and a boundary pair; this establishes the connection with the Kre˘ın–Birman–Vishik theory. The general theory is applied to boundary value problems for differential operators in Chapters 6–8 in three different situations. In each case the presentation follows a similar scheme: After the necessary preparations to keep these chapters mostly self-contained, explicit boundary triplets and Weyl functions for the particular operators or relations under consideration, are provided. A further spectral analysis, depending on the nature of problem is presented. The class of Sturm–Liouville operators that is discussed in Chapter 6 covers also the example given earlier in this introduction. A good deal of preparation is needed to construct closed semibounded forms and corresponding boundary pairs in the singular situation. Chapter 7 deals with 2 × 2 canonical systems of differential equations and also illustrates the role of linear relations in the analysis of such systems. Finally, in Chapter 8 Schr¨ odinger operators on bounded domains Ω ⊂ Rn are treated, where one of the main challenges is to construct Dirichlet and Neumann traces on the maximal domain..

(19) 10. Introduction. For the reader’s convenience a number of appendices have been added: they contain material concerning Nevanlinna functions and some useful elementary observations on operators and subspaces in Hilbert spaces. At the end of the text a few notes and some (historical) comments, as well as a list of recent and earlier references, can be found. Here the reader is also referred to some recent literature for topics that go beyond this monograph.. Open Access This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made. The images or other third party material in this chapter are included in the chapter’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder..

(20) Chapter 1. Linear Relations in Hilbert Spaces A linear relation from one Hilbert space to another Hilbert space is a linear subspace of the product of these spaces. In this chapter some material about such linear relations is presented and it is shown how linear operators, whether densely defined or not, fit in this context. The basic terminology is provided in Section 1.1 and afterwards the spectrum, resolvent set, the adjoint, and operator decompositions of linear relations are discussed in Section 1.2 and Section 1.3. Linear relations with special properties, such as symmetric, self-adjoint, dissipative, and accumulative relations, are investigated in Sections 1.4, 1.5, and 1.6. More details on self-adjoint and semibounded relations can be found in Chapter 3 and Chapter 5. Intermediate extensions and the classical von Neumann formulas describing self-adjoint extensions of symmetric operators and relations can be found in Section 1.7. In Section 1.8 it is shown that there is a natural indefinite inner product by means of which the notion of adjoint relation corresponds to the notion of orthogonal companion. Strong graph convergence and strong resolvent convergence of sequences of linear relations are discussed in Section 1.9 and parametric representations of linear relations are studied in Section 1.10. Finally, in Section 1.11 some useful properties of a resolvent-type operator of a linear relation are given, and in Section 1.12 the class of so-called Nevanlinna families, a natural extension of the class of Nevanlinna functions (see Appendix A) is studied.. 1.1. Elementary facts about linear relations. Let H and K be Hilbert spaces over C. The Hilbert space inner product and the corresponding norm are usually denoted by (·, ·) and · , respectively, and sometimes a subindex will be used in order to avoid confusion. The inner product is linear in the first entry and antilinear in the second entry. The orthogonal complement will be denoted by ⊥, sometimes a subindex will be used to indicate the relevant space. The product H × K will often be regarded as a Hilbert space with the standard inner product (·, ·)H + (·, ·)K and all topological notions in H × K. © The Author(s) 2020 J. Behrndt et al., Boundary Value Problems, Weyl Functions, and Differential Operators, Monographs in Mathematics 108, https://doi.org/10.1007/978-3-030-36714-5_2. 11.

(21) 12. Chapter 1. Linear Relations in Hilbert Spaces. are understood with respect to the topology induced by the corresponding norm. The product space H × K will also be written as H ⊕ K, and H and K are then regarded as closed linear subspaces in H ⊕ K which are orthogonal to each other. A linear subspace of H × K is called a linear relation from H to K. If H is a linear relation from H to K the elements h ∈ H will in general be written as pairs {h, h } with components h ∈ H and h ∈ K. If K = H one speaks simply of a linear relation in H. After this introductory section the adjective linear is usually omitted and one speaks of relations when linear relations are meant. The domain, range, kernel, and multivalued part of a linear relation H from H to K are defined by dom H = h ∈ H : {h, h } ∈ H for some h ∈ K , ran H = h ∈ K : {h, h } ∈ H for some h ∈ H , ker H = h ∈ H : {h, 0} ∈ H , mul H = h ∈ K : {0, h } ∈ H , respectively. The closure of the linear space dom H will be denoted by dom H and, likewise, the closure of the linear space ran H will be denoted by ran H. Note that each linear operator H from H to K is a linear relation if the operator is identified with its graph, H = {h, Hh} : h ∈ dom H , and that a linear relation H is (the graph of) an operator if and only if the multivalued part of H is trivial, mul H = {0}. The inverse H −1 of a linear relation H from H to K is defined by H −1 = {h , h} : {h, h } ∈ H , so that H −1 is a linear relation from K to H. In the next lemma some obvious facts concerning the inverse relation are collected. Lemma 1.1.1. Let H be a linear relation from H to K. Then the following identities hold: dom H −1 = ran H, ker H −1 = mul H,. ran H −1 = dom H, mul H −1 = ker H.. There is a linear structure on the collection of linear relations from H to K. For linear relations H and K from H to K the componentwise sum is the linear relation from H to K defined by K = {h + k, h + k } : {h, h } ∈ H, {k, k } ∈ K , (1.1.1) H + while the product λH of H with a scalar λ ∈ C is the linear relation from H to K defined by λH = {h, λh } : {h, h } ∈ H ..

(22) 1.1. Elementary facts about linear relations. 13. K is the linear span of the graphs of H Note that the componentwise sum H + and K, and K) = dom H + dom K, dom (H +. K) = ran H + ran K. ran (H +. Likewise, if λ ∈ C, one has dom λH = dom H. and for. λ = 0. ran λH = ran H.. Note that by definition 0 H = Odom H , where Odom H stands for the zero operator on dom H. It is useful to note that K)−1 = H −1 + K −1 , (H +. (λH)−1 =. 1 −1 H , λ. λ = 0.. Let H and K be linear relations from H to K. If H ⊂ K, then H is called a restriction of K and K is an extension of H. Proposition 1.1.2. Let H and K be linear relations from H to K and assume that H ⊂ K. Then {0} × mul K , dom H = dom K ⇔ K = H + (1.1.2) and, analogously, ran H = ran K. ⇔. ker K + {0} . K=H +. (1.1.3). Proof. Note that H ⊂ K is equivalent to H −1 ⊂ K −1 . Hence, in order to prove (1.1.3) one just applies (1.1.2) with H and K replaced by H −1 and K −1 , respectively. Thus it suffices to show (1.1.2). The implication (⇐) is trivial. To show ({0} × mul K) ⊂ K and hence it suffices to (⇒), observe that H ⊂ K yields H + ({0} × mul K). Let {h, h } ∈ K. Since h ∈ dom K = dom H, show that K ⊂ H + there exists an element k ∈ K such that {h, k } ∈ H and from H ⊂ K it follows that also {h, k } ∈ K. Hence, with ϕ = h − k one has {h, h } = {h, k } + {0, ϕ }, and thus {0, ϕ } ∈ K, i.e., ϕ ∈ mul K.. . Corollary 1.1.3. Let H and K be linear relations from H to K and assume that H ⊂ K. Then dom H = dom K. and. mul H = mul K. ⇔. H = K,. (1.1.4). and, analogously, ran H = ran K. and. ker H = ker K. ⇔. H = K.. (1.1.5).

(23) 14. Chapter 1. Linear Relations in Hilbert Spaces. Proof. It suffices to show (1.1.4), as (1.1.5) follows by taking inverses in (1.1.4). Clearly, the implication (⇐) is trivial. For the implication (⇒) apply (1.1.2). Then dom H = dom K and mul H = mul K give successively {0} × mul K = H + {0} × mul H ⊂ H, K=H + which together with H ⊂ K implies H = K.. . Let H and K be linear relations from H to K. The usual (operatorwise) sum H + K is defined by H + K = {h, h + h } : {h, h } ∈ H, {h, h } ∈ K , where dom (H + K) = dom H ∩ dom K. Note that mul (H + K) = mul H + mul K. If H is a linear relation in H, then for λ ∈ C the sum H + λI, where I denotes the identity operator in H, is usually simply written as H + λ and has the form H + λ = {h, h + λh} : {h, h } ∈ H , with dom (H + λ) = dom H. Note that mul (H + λ) = mul H. Let H be a linear relation from H to K and let K be a linear relation from K to G, where G is another Hilbert space. Then the product KH of K and H is the linear relation from H to G defined by KH = {h, h } : {h, h } ∈ H, {h , h } ∈ K . Note that for λ ∈ C the notation λH agrees with (λI)H, where I denotes the identity operator in K. It is straightforward to check that (KH)−1 = H −1 K −1 . The following lemma shows an important feature of sums and products of linear relations. The notation IM stands for the identity operator on the linear subspace M, while OM stands for the zero operator on M. Lemma 1.1.4. Let H be a linear relation from H to K. Then {0} × mul H , H + (−H) = Odom H +. (1.1.6). where the sum is direct. Moreover, the identities {0} × mul H HH −1 = Iran H +. (1.1.7). and. {0} × ker H H −1 H = Idom H +. (1.1.8). hold, and both sums are direct. Proof. First the identity (1.1.6) will be shown. For an element on the left-hand side of (1.1.6) one has {h, h − h } = {h, 0} + {0, h − h }, where {h, h }, {h, h } ∈ H, so that {h, 0} ∈ Odom H and {0, h −h } ∈ {0}×mul H..

(24) 1.1. Elementary facts about linear relations. 15. ({0}×mul H). Then {h, k} = {h, 0}+{0, k} Conversely, let {h, k} ∈ Odom H + with h ∈ dom H and k ∈ mul H. Hence, {h, h } ∈ H for some h ∈ K so that also {h, h − k} ∈ H. Consequently, {h, k} = {h, h − (h − k)} ∈ H + (−H), which completes the proof of (1.1.6). The assertion (1.1.8) follows from (1.1.7) by replacing H with H −1 . Hence, only the identity in (1.1.7) has to be proved. By definition, the linear relation HH −1 is given by HH −1 = {h, h } : {h, h } ∈ H −1 , {h , h } ∈ H . Therefore, if {h, h } ∈ HH −1 with some {h, h } ∈ H −1 and {h , h } ∈ H, then {h, h } = {h, h} + {0, h − h}. As {h , h} ∈ H, it follows that h ∈ ran H and {0, h − h} = {h , h } − {h , h} ∈ H, ({0} × mul H). i.e., h − h ∈ mul H. Thus, {h, h } ∈ Iran H + ({0} × mul H) with Conversely, given an element {h, h} + {0, k} ∈ Iran H + h ∈ ran H and k ∈ mul H, there exists h ∈ dom H such that {h , h} ∈ H or, equivalently, {h, h } ∈ H −1 . Since {0, k} ∈ H it follows {h , h + k} ∈ H, so that {h, h + k} ∈ HH −1 . Thus far the Hilbert space structure of the spaces has not been used; only the linear space structure played a role. Now an interpretation of the componentwise K in (1.1.1) will be given as an orthogonal componentwise sum. Let H1 , sum H + H2 , K1 , and K2 be Hilbert spaces and let H = H1 ⊕ H2 and K = K1 ⊕ K2 . Here and in the following H1 and H2 are viewed as closed linear subspaces of H, and K1 and K2 are viewed as closed linear subspaces of K. Assume that H is a linear relation from H1 to K1 and that K is a linear relation from H2 to K2 . The orthogonal sum K is defined as H ⊕ K = {h + k, h + k } : {h, h } ∈ H, {k, k } ∈ K . H ⊕ K is just the componentwise sum H + K of H and K, when In other words, H ⊕ these linear relations H and K are interpreted as linear relations from H = H1 ⊕H2 to K = K1 ⊕ K2 . If H = K and H1 = K1 , H2 = K2 , then this definition implies K)2 = H 2 ⊕ K 2. (H ⊕. (1.1.9). A linear relation H from H to K is called bounded if there is a constant C ≥ 0 such that h K ≤ C h H for all {h, h } ∈ H. In this case it is clear that.

(25) 16. Chapter 1. Linear Relations in Hilbert Spaces. mul H = {0}, so that H is a bounded operator. Thus, there is no distinction between bounded linear relations or bounded linear operators. The set of everywhere defined bounded linear operators from H to K will be denoted by B(H, K). If H = K the notation B(H) is used instead of B(H, H). A linear relation from H to K is called closed if it is closed as a linear subspace of H × K. The closure H of the linear relation H as a linear subspace of H × K is itself a closed linear relation. It follows that mul H ⊂ mul H; if mul H = {0} implies that mul H = {0}, then the operator H is called closable (as an operator). The following useful observations are easily verified. Lemma 1.1.5. Let H be a linear operator from H to K. Then the following statements hold: (i) Let H be closable. If dom H is closed, then H is closed. (ii) Let H be bounded. Then H is closable. (iii) Let H be bounded. Then dom H is closed if and only if H is closed. A linear relation H from H to K is called contractive if h K ≤ h H for all {h, h } ∈ H and it is called isometric if h K = h H for all {h, h } ∈ H. In each case mul H = {0} and H is an operator which is bounded and thus closable; cf. Lemma 1.1.5. Hence, there is no distinction between contractive relations or operators. Likewise, there is no distinction between isometric relations or operators. Clearly, the closure of a contractive or isometric operator is again contractive or isometric. Recall that a contraction H has the following useful property: if. Hk K = k H for some k ∈ dom H, then (Hh, Hk)K = (h, k)H. for all. h ∈ dom H.. (1.1.10). To see this, note that for all λ ∈ C 0 ≤ h + λk 2H − H(h + λk) 2K = h 2H − Hh 2K − 2Re λ [(Hh, Hk)K − (h, k)H ] , which implies that (1.1.10) holds. For many combinations of linear relations the closedness is preserved. For instance, if H is a closed linear relation from H to K, then H −1 is a closed linear relation from K to H. Likewise, for λ = 0 the product λH is closed. If H and K K is are closed linear relations from H to K, then the componentwise sum H + not necessarily closed (see Appendix C), while the orthogonal componentwise sum K of H and K is closed. The sum H + K of two closed linear relations H H ⊕ and K is not necessarily closed. However, in the special case that H is closed and K ∈ B(H, K) the sum H + K = {h, h + Kh} : {h, h } ∈ H is also closed. In particular, the linear relation H in H is closed if and only if H + λ is closed for some, and hence for all λ ∈ C. The product KH of closed linear.

(26) 1.1. Elementary facts about linear relations. 17. relations K and H is not necessarily closed. However, in the special case that K is closed and H ∈ B(H, K) the product KH = {h, h } : {Hh, h } ∈ K is also closed. The above material will be used throughout the text. The rest of this section will be devoted to two specific items, namely, a discussion of questions around the so-called resolvent identity, and one involving Möbius transformations of linear relations. For a linear relation H in H and λ ∈ C, the resolvent relation is defined by (H − λ)−1 . Clearly, H is closed if and only if (H − λ)−1 is closed for some, and hence for all λ ∈ C. The resolvent relation has a number of properties which will now be explored. First the λ-independence of ker (H − λ)−1 and mul (H − λ) is stated. Lemma 1.1.6. Let H be a linear relation in H and let λ ∈ C. Then ker (H − λ)−1 = mul (H − λ) = mul H. For practical purposes it is worthwhile mentioning the analogs of (1.1.7) and (1.1.8) for the resolvent relation of H. Using Lemma 1.1.6 one sees that {0} × mul H , (H − λ)(H − λ)−1 = Iran (H−λ) + and, likewise, {0} × ker (H − λ) . (H − λ)−1 (H − λ) = Idom H + In particular, when ker (H − λ) = {0} for some λ ∈ C, one has (H − λ)−1 (H − λ) = Idom H . The resolvent identity in the next proposition involves a combination of the sum and the product of the resolvent relations (H − λ)−1 and (H − μ)−1 . Proposition 1.1.7. Let H be a linear relation in H and let λ, μ ∈ C. Then (H − λ)−1 − (H − μ)−1 = (H − λ)−1 (λ − μ)(H − μ)−1 .. (1.1.11). If ker (H − λ) = {0} and ker (H − μ) = {0}, then (H − λ)−1 and (H − μ)−1 are linear operators defined on ran (H −λ) and ran (H −μ), respectively, with the same kernel mul H. Moreover, if λ = μ, then (1.1.11) may be written as (H − λ)−1 − (H − μ)−1 = (λ − μ)(H − λ)−1 (H − μ)−1 .. (1.1.12).

(27) 18. Chapter 1. Linear Relations in Hilbert Spaces. Proof. For the inclusion (⊂) in (1.1.11) let {h, h − h } ∈ (H − λ)−1 − (H − μ)−1 , with {h, h } ∈ (H − λ)−1 and {h, h } ∈ (H − μ)−1 . This gives {h , h + λh } ∈ H. and. {h , h + μh } ∈ H,. which shows {h − h , λh − μh } ∈ H, and thus {h − h , (λ − μ)h } ∈ H − λ and (λ − μ)h , h − h ∈ (H − λ)−1 . Since {h, h } ∈ (H − μ)−1 , one sees that {h, (λ − μ)h } ∈ (λ − μ)(H − μ)−1 , as {h , (λ − μ)h } ∈ (λ − μ)I. Hence, the element {h, h − h } belongs to the linear relation (H − λ)−1 (λ − μ)(H − μ)−1 , which shows the inclusion. For the inclusion (⊃) in (1.1.11), let {h, h } ∈ (H − λ)−1 (λ − μ)(H − μ)−1 . Then by definition there exists k ∈ H such that {h, k} ∈ (H − μ)−1. and. {(λ − μ)k, h } ∈ (H − λ)−1 ,. as {k, (λ − μ)k} ∈ (λ − μ)I. In addition, it is clear from {k, h} ∈ H − μ that {k, h + (μ − λ)k} ∈ H − λ and {h + (μ − λ)k, k} ∈ (H − λ)−1 . Thus, it follows that {h, h + k} ∈ (H − λ)−1 . Hence, {h, h } = {h, h + k − k} belongs to (H − λ)−1 − (H − μ)−1 , which shows the inclusion. This completes the proof of (1.1.11). If λ = μ this leads to (1.1.12). The remaining statements follow directly from Lemma 1.1.6. Note that in general the identity in (1.1.12) is not valid for λ = μ. In this case the right-hand side of (1.1.12) clearly equals Odom (H−λ)−2 , while by (1.1.6) ({0} × mul (H − λ)−1 ). Hence, in (1.1.12) the left-hand side equals Odom (H−λ)−1 + the right-hand side is contained in the left-hand side. The following result shows that every linear relation H can be represented by means of a pair of operators expressed in terms of its resolvent operator (H −λ)−1 . This kind of representation of a linear relation will be considered in this text in various situations. Lemma 1.1.8. Let H be a linear relation in H and assume that ker (H − λ) = {0} for some λ ∈ C. Then (1.1.13) H = {(H − λ)−1 k, (I + λ(H − λ)−1 )k} : k ∈ ran (H − λ) , where the right-hand side is well defined since dom (H − λ)−1 = ran (H − λ)..

(28) 1.1. Elementary facts about linear relations. 19. Proof. Denote the linear relation on the right-hand side of (1.1.13) by K. To see that H ⊂ K, let {h, h } ∈ H. Then {h − λh, h} ∈ (H − λ)−1 and from the assumption mul (H − λ)−1 = ker (H − λ) = {0} it follows that h = (H − λ)−1 (h − λh). Therefore, {h, h } = h, h − λh + λh = (H − λ)−1 (h − λh), (I + λ(H − λ)−1 )(h − λh) , where h −λh ∈ ran (H −λ). Hence, {h, h } ∈ K, so that H ⊂ K. Now the equality follows from Corollary 1.1.3, since dom K = ran (H − λ)−1 = dom (H − λ) = dom H, while. mul K = ker (H − λ)−1 = mul (H − λ) = mul H.. This completes the proof.. −1. Another algebraic identity involving the resolvent relations (H − λ) and (H − μ)−1 is contained in the next lemma; see also Corollary 1.2.8 in the next section. The formula in the lemma can also be checked via the Möbius transform to be defined below. Lemma 1.1.9. Let H be a linear relation in H and let λ, μ ∈ C. Then −1 (I + (λ − μ)(H − λ)−1 = I + (μ − λ)(H − μ)−1 .. (1.1.14). Proof. It is easy to see that I + (λ − μ)(H − λ)−1 = {h − λh, h − μh} : {h, h } ∈ H , and by symmetry I + (μ − λ)(H − μ)−1 = {h − μh, h − λh} : {h, h } ∈ H . This yields (1.1.14).. . Next, M¨ obius transformations of linear relations will be defined. For a Hilbert space H and a 2 × 2 matrix . α β M= , α, β, γ, δ ∈ C, (1.1.15) γ δ the scalar M¨ obius transform M in H2 = H × H is given by M : H2 → H2 ,. {h, h } → {αh + βh , γh + δh }.. The meaning of M, either as a matrix or as a transformation, will be clear from the context. The scalar M¨ obius transform of a linear relation is defined as follows..

(29) 20. Chapter 1. Linear Relations in Hilbert Spaces. Definition 1.1.10. Let H be a linear relation in H and let M be a 2 × 2 matrix as in (1.1.15). Then the scalar M¨ obius transform of H is the linear relation M[H] in H defined by (1.1.16) M[H] = {αh + βh , γh + δh } : {h, h } ∈ H . Note that the domain and range of the scalar Möbius transform M[H] are given by dom M[H] = αh + βh : {h, h } ∈ H , ran M[H] = γh + δh : {h, h } ∈ H . If the 2 × 2 matrix M in Definition 1.1.10 is multiplied by a constant η ∈ C \ {0}, then the corresponding M¨ obius transform M[H] and (ηM)[H] coincide. Let M and N be 2 × 2 matrices. Then the identity N[M[H]] = (N ◦ M)[H]. (1.1.17). holds for any linear relation H in H. If det M = 0, then . 1 δ −β −1 M = αδ − βγ −γ α and the M¨ obius transform corresponding to M−1 is given by M−1 [H] = {δh − βh , −γh + αh } : {h, h } ∈ H . Thus, for any linear relation H one has M−1 [M[H]] = H = M[M−1 [H]]; cf. (1.1.17). Note that in general M−1 [H] and M[H]−1 are different relations. In the case det M = 0 it clearly follows that M[H] is closed if and only if H is closed.. (1.1.18). Observe that the linear relations λH, H − λ, H −1 correspond to the M¨ obius transforms determined by the following matrices . . . 1 0 1 0 0 1 , , , 0 λ −λ 1 1 0 respectively. Thus, for instance, the linear relations I + (λ − μ)(H − λ)−1 and I + (μ − λ)(H − μ)−1 correspond to M¨ obius transforms of H determined by the matrices . 1 0 1 0 0 1 1 0 −λ 1 = 1 1 0 λ−μ 1 0 −λ 1 −μ 1.

(30) 1.1. Elementary facts about linear relations. and. 1 1. . 0 1. 1 0. . 0 μ−λ. 0 1. 21. . 1 0. 1 −μ. 0 1. −μ −λ. =. 1 , 1. respectively. This also confirms the identity (1.1.14). For a 2 × 2 matrix M as in (1.1.15) with det M = 0 define the function λ → M[λ] =. γ + λδ , α + λβ. α + λβ = 0.. (1.1.19). Since the linear relation M[H] − M[λ] corresponds to the matrix. α β 1 0 α β = −λ det M det M , −M[λ] 1 γ δ α+λβ α+λβ one sees from (1.1.16) that for α + λβ = 0,. det M (h − λh) : {h, h } ∈ H . M[H] − M[λ] = αh + βh , α+λβ This identity yields, in particular, for α + βλ = 0, that ker (H − λ) = ker M[H] − M[λ] , ran (H − λ) = ran M[H] − M[λ] .. (1.1.20). If, in addition, β = 0, then it follows from (1.1.16) that mul M[H] = ker (H + αβ −1 ),. mul H = ker M[H] − δβ −1 ,. and in the case β = 0 it is easy to see that mul M[H] = mul H. Proposition 1.1.11. Let H be a linear relation in H and let M be a 2 × 2 matrix as in (1.1.15) with det M = 0. Then for α + λβ = 0 −1 (α + λβ)2 (α + λβ)β M[H] − M[λ] + (H − λ)−1 . = det M det M. (1.1.21). Proof. Use the abbreviation Δ = det M. It suffices to see that the left-hand side corresponds to the matrix −λΔ. Δ 0 1 1 0 α β = α+λβ α+λβ , 1 0 −M[λ] 1 γ δ α β while the right-hand side corresponds to the matrix

(31) 1 0 1 0 −λ 0 1 1 0 2 = α(α+λβ) (α+λβ)β (α+λβ) 1 0 −λ 1 1 0 Δ Δ Δ. 1 β(α+λβ) Δ. .. Since these matrices coincide up to a nonzero multiplicative constant the assertion follows. .

(32) 22. Chapter 1. Linear Relations in Hilbert Spaces. It is clear that the following useful consequence of Proposition 1.1.11 is obtained by means of the special choice . 0 1 M= , 1 0 so that det M = −1, M[H] = H −1 , and M[λ] = 1/λ, λ = 0. Corollary 1.1.12. Let H be a linear relation in H and let λ ∈ C \ {0}. Then (H −1 − λ−1 )−1 = −λ − λ2 (H − λ). −1. .. (1.1.22). Next the Cayley transform and inverse Cayley transform of a linear relation will be introduced. These special M¨ obius transforms will be used later in Sections 1.5, 1.6, and 1.7. Definition 1.1.13. Let H and V be linear relations in H and let μ ∈ C \ R. Then the Cayley transform Cμ of H and the inverse Cayley transform Fμ of V are defined by Cμ [H] = {h − μh, h − μh} : {h, h } ∈ H , (1.1.23) Fμ [V ] = {k − k , μk − μk } : {k, k } ∈ V . Notice that the domain and range of the Cayley transform Cμ and the inverse Cayley transform Fμ are given by dom Cμ [H] = ran (H − μ), dom Fμ [V ] = ran (I − V ),. ran Cμ [H] = ran (H − μ), ran Fμ [V ] = ran (μ − μV ).. (1.1.24). It is clear that the Cayley transform Cμ and the inverse Cayley transform Fμ are M¨ obius transforms corresponding to the matrices. −μ 1 1 −1 and Fμ = = (μ − μ)C−1 (1.1.25) Cμ = μ , −μ 1 μ −μ where det Cμ = μ − μ was used. Note also that Cμ [λ] =. λ−μ , λ−μ. λ = μ.. Thus, Proposition 1.1.11 leads to the following result. Corollary 1.1.14. Let H be a linear relation in H and let μ ∈ C \ R. Then . Cμ [H] − Cμ [λ]. −1. =. λ − μ (λ − μ)2 + (H − λ)−1 , μ−μ μ−μ. λ = μ.. (1.1.26).

(33) 1.2. Spectra, resolvent sets, and points of regular type. 1.2. 23. Spectra, resolvent sets, and points of regular type. The resolvent set, spectrum, point, continuous, and residual spectrum, and the points of regular type of a linear relation or operator are defined. A priori it is not assumed that the linear relation is closed. Here and in the rest of the text linear relations will be referred to simply as relations and linear subspaces as subspaces. Definition 1.2.1. Let H be a relation in H. Then λ ∈ C is said to be a point of regular type of H if (H − λ)−1 is a (in general not everywhere defined) bounded operator. The set of points of regular type of H is denoted by γ(H). Some straightforward consequences of Definition 1.2.1 are presented in the next lemma. Lemma 1.2.2. Let H be a relation in H. Then λ ∈ γ(H) if and only if there exists a positive constant c, depending on λ, such that. h ≤ c h − λh ,. {h, h } ∈ H.. (1.2.1). Moreover, if γ(H) = ∅, then H is closed if and only if ran (H − λ) is closed for some, and hence for all λ ∈ γ(H). Proof. Assume that λ ∈ γ(H), so that (H − λ)−1 is a bounded operator. Let {h, h } ∈ H; then {h − λh, h} ∈ (H − λ)−1 and. h = (H − λ)−1 (h − λh) ≤ c h − λh , which gives (1.2.1). Conversely, assume that (1.2.1) holds. To see that (H − λ)−1 is a bounded operator let {f, f } ∈ (H − λ)−1 . Then {f, f } = {h − λh, h} for some {h, h } ∈ H and (1.2.1) shows f ≤ c f for all {f, f } ∈ (H − λ)−1 . This implies that (H − λ)−1 is an operator that is bounded or, equivalently, λ ∈ γ(H). Assume that H is closed, so that also (H − λ)−1 is closed. Then the relation (H − λ)−1 is a closed and bounded operator for all λ ∈ γ(H). This immediately implies that ran (H − λ) = dom (H − λ)−1 is closed; cf. Lemma 1.1.5. Conversely, if ran (H − λ) = dom (H − λ)−1 is closed for some λ ∈ γ(H), then (H − λ)−1 is a bounded operator defined on a closed subspace. It follows that (H − λ)−1 is closed, cf. Lemma 1.1.5, and hence H is closed. Definition 1.2.3. Let H be a relation in H. A point λ ∈ C is said to belong to the resolvent set ρ(H) of H if (H − λ)−1 is a bounded operator and ran (H − λ) = H. The spectrum σ(H) of H is the complement of ρ(H) in C. The spectrum σ(H) decomposes into three disjoint components: the point spectrum σp (H), continuous spectrum σc (H), and residual spectrum σr (H), defined by σp (H) = λ ∈ C : ker (H − λ) = {0} , σc (H) = λ ∈ C : ker (H − λ) = {0}, ran (H − λ) = H, λ ∈ ρ(H) , σr (H) = λ ∈ C : ker (H − λ) = {0}, ran (H − λ) = H ..

(34) 24. Chapter 1. Linear Relations in Hilbert Spaces. Let H be a relation in H. It follows from Definition 1.2.1 and Definition 1.2.3 that ρ(H) ⊂ γ(H). Moreover, it follows from (1.2.1) that γ(H) = γ(H) and the equivalence ran (H − λ) = H ⇔ ran (H − λ) = H implies ρ(H) = ρ(H). The following state diagram is useful when discussing the spectral subsets and the resolvent set of H. The top row shows all possibilities for the range of H −λ. The first (second) rows show all possibilities for points λ such that (H −λ)−1 is a bounded (unbounded) operator and the bottom row shows all possibilities for eigenvalues λ. ran (H − λ) =H. ran (H − λ) dense, = H. ran (H − λ) not dense. (H − λ)−1 bounded operator. ρ(H). ρ(H). γ(H) ∩ σr (H). (H − λ)−1 unbounded operator. σc (H). σc (H). σr (H). (H − λ)−1 not operator. σp (H). σp (H). σp (H). Now assume that H is a closed relation. Then it follows with the help of the closed graph theorem and Lemma 1.1.5 applied to the operator (H − λ)−1 that two cases (marked by X below) in the above state diagram are not possible: H closed relation. ran (H − λ) =H. ran (H − λ) dense, = H. ran (H − λ) not dense. (H − λ)−1 bounded operator. ρ(H). X. γ(H) ∩ σr (H). (H − λ)−1 unbounded operator. X. σc (H). σr (H). (H − λ)−1 not operator. σp (H). σp (H). σp (H). In particular, for a closed relation the continuous spectrum is given by σc (H) = λ ∈ C : ker (H − λ) = {0}, ran (H − λ) = H, ran (H − λ) = H . Lemma 1.2.4. A relation H in H is closed if and only if ran (H − λ) = H for some, and hence for all λ ∈ ρ(H). In this case the following statements hold:.

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