Graduate Texts in Mathematics 265

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Graduate Texts in Mathematics 265

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Graduate Texts in Mathematics

Series Editors:

Sheldon Axler

San Francisco State University, San Francisco, CA, USA Kenneth Ribet

University of California, Berkeley, CA, USA

Advisory Board:

Colin Adams, Williams College, Williamstown, MA, USA

Alejandro Adem, University of British Columbia, Vancouver, BC, Canada Ruth Charney, Brandeis University, Waltham, MA, USA

Irene M. Gamba, The University of Texas at Austin, Austin, TX, USA Roger E. Howe, Yale University, New Haven, CT, USA

David Jerison, Massachusetts Institute of Technology, Cambridge, MA, USA Jeffrey C. Lagarias, University of Michigan, Ann Arbor, MI, USA

Jill Pipher, Brown University, Providence, RI, USA

Fadil Santosa, University of Minnesota, Minneapolis, MN, USA Amie Wilkinson, University of Chicago, Chicago, IL, USA

Graduate Texts in Mathematics bridge the gap between passive study and creative understanding, offering graduate-level introductions to advanced topics in mathematics. The volumes are carefully written as teaching aids and highlight characteristic features of the theory. Although these books are frequently used as textbooks in graduate courses, they are also suitable for individual study.

For further volumes:

www.springer.com/series/136

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Konrad Schmüdgen

Unbounded

Self-adjoint Operators

on Hilbert Space

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Dept. of Mathematics University of Leipzig Leipzig, Germany

ISSN 0072-5285 Graduate Texts in Mathematics

ISBN 978-94-007-4752-4 ISBN 978-94-007-4753-1 (eBook) DOI 10.1007/978-94-007-4753-1

Springer Dordrecht Heidelberg New York London

Library of Congress Control Number: 2012942602

Mathematics Subject Classification: 47B25, 81Q10 (Primary), 47A, 47E05, 47F05, 35J (Secondary)

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To ELISA

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Preface and Overview

This book is designed as an advanced text on unbounded self-adjoint operators in Hilbert space and their spectral theory, with an emphasis on applications in mathematical physics and various fields of mathematics. Though in several sections other classes of unbounded operators (normal operators, symmetric operators, accretive operators, sectorial operators) appear in a natural way and are developed, the leit- motif is the class of unbounded self-adjoint operators.

Before we turn to the aims and the contents of this book, we briefly explain the two main notions occurring therein. Suppose thatHis a Hilbert space with scalar product·,·andT is a linear operator onHdefined on a dense linear subspace D(T ). ThenT is said to be it symmetric if

T x, y = x, T y forx, y∈D(T ). (0.1) The operatorT is called self-adjoint if it is symmetric and if the following property is satisfied: Suppose thaty∈Hand there exists a vectoru∈Hsuch thatT x, y = x, ufor allx∈D(T ). Theny lies inD(T ). (SinceD(T )is assumed to be dense inH, it follows then thatu=T y.)

Usually it is easy to verify that Eq. (0.1) holds, so that the corresponding operator is symmetric. For instance, ifΩ is an open bounded subset ofR^d andT is the LaplacianΔonD(T )=C₀^∞(Ω)in the Hilbert spaceL²(Ω), a simple integration- by-parts computation yields (0.1). If the symmetric operator is bounded, its continuous extension toHis self-adjoint. However, for unbounded operators, it is often difficult to prove or not true (as in the caseT =Δ) that a symmetric operator is self- adjoint. Differential operators and most operators occurring in mathematical physics are not bounded. Dealing with unbounded operators leads not only to many techni- cal subtleties; it often requires the development of new methods or the invention of new concepts.

Self-adjoint operators are fundamental objects in mathematics and in quantum physics. The spectral theorem states that any self-adjoint operatorT has an integral representationT =

λ dE(λ)with respect to some unique spectral measureE. This gives the possibility to define functionsf (T )=

f (λ) dE(λ) of the operator and to develop a functional calculus as a powerful tool for applications. The spectrum vii

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of a self-adjoint operator is always a subset of the reals. In quantum physics it is postulated that each observable is given by a self-adjoint operatorT. The spectrum ofT is then the set of possible measured values of the observable, and for any unit vectorx∈Hand subsetM⊆R, the numberE(M)x, xis the probability that the measured value in the statex lies in the setM. IfT is the Hamilton operator, the one-parameter unitary groupt→e^{it T} describes the quantum dynamics.

All this requires the operatorT to be self-adjoint. For general symmetric oper- atorsT, the spectrum is no longer a subset of the reals, and it is impossible to get an integral representationT =

λ dE(λ)or to define the exponentiatione^{it T}. That is, the distinction between symmetric operators and self-adjoint operators is crucial!

However, many symmetric operators that are not self-adjoint can be extended to a self-adjoint operator acting on the same Hilbert space.

The main aims of this book are the following:

• to provide a detailed study of unbounded self-adjoint operators and their properties,

• to develop methods for proving the self-adjointness of symmetric operators,

• to study and describe self-adjoint extensions of symmetric operators.

A particular focus and careful consideration is on the technical subtleties and difficulties that arise when dealing with unbounded operators.

Let us give an overview of the contents of the book. Part I is concerned with the basics of unbounded closed operators on a Hilbert space. These include fundamental general concepts such as regular points, defect numbers, spectrum and resolvent, and classes of operators such as symmetric and self-adjoint operators, accretive and sectorial operators, and normal operators.

Our first main goal is the theory of spectral integrals and the spectral decomposition of self-adjoint and normal operators, which is treated in detail in Part II. We use the bounded transform to reduce the case of unbounded operators to bounded ones and derive the spectral theorem in great generality for finitely many strongly commuting unbounded normal operators. The functional calculus for self-adjoint operators developed here will be will be essential for the remainder of the book.

Part III deals with generators of one-parameter groups and semigroups, as well as with a number of important and technical topics including the polar decomposition, quasi-analytic and analytic vectors, and tensor products of unbounded operators.

The second main theme of the book, addressed in Part IV, is perturbations of self- adjointness and of spectra of self-adjoint operators. The Kato–Rellich theorem, the invariance of the essential spectrum under compact perturbations, the Aronszajn–

Donoghue theory of rank one perturbations and Krein’s spectral shift and trace formula are treated therein. A guiding motivation for many results in the book, and in this part in particular, are applications to Schrödinger operators arising in quantum mechanics.

Part V contains a detailed and concise presentation of the theory of forms and their associated operators. This is the third main theme of the book. Here the cen- tral results are three representation theorems for closed forms, one for lower semi- bounded Hermitian forms and two others for bounded coercive forms and for sectorial forms. Other topics treated include the Friedrichs extension, the order relation

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Preface and Overview ix of self-adjoint operators, and the min–max principle. The results on forms are applied to the study of differential operators. The Dirichlet and Neumann Laplacians on bounded open subsets ofR^d and Weyl’s asymptotic formula for the eigenvalues of the Dirichlet Laplacian are developed in detail.

The fourth major main theme of the book, featured in Part VI, is the self- adjoint extension theory of symmetric operators. First, von Neumann’s theory of self-adjoint extensions, and Krein’s theory and the Ando–Nishio theorem on positive self-adjoint extensions are investigated. The second chapter in Part VI gives an extensive presentation of the theory of boundary triplets. The Krein–Naimark resolvent formula and the Krein–Birman–Vishik theory on positive self-adjoint extensions are treated in this context. The two last chapters of Part VI are concerned with two important topics where self-adjointness and self-adjoint extensions play a crucial role. These are Sturm–Liouville operators and the Hamburger moment prob- lem on the real line.

Throughout the book applications to Schrödinger operators and differential operators are our guiding motivation, and while a number of special operator-theoretic results on these operators are presented, it is worth stating that this is not a research monograph on such operators. Again, the emphasis is on the general theory of un- bounded self-adjoint Hilbert space operators. Consequently, basic definitions and facts on such topics as Sobolev spaces are collected in an appendix; whenever they are needed for applications to differential operators, they are taken for granted.

This book is an outgrowth of courses on various topics around the theory of unbounded self-adjoint operators and their applications, given for graduate and Ph.D. students over the past several decades at the University of Leipzig. Some of these covered advanced topics, where the material was mainly to be found in research papers and monographs, with any suitable advanced text notably missing.

Most chapters of this book are drawn from these lectures. I have tried to keep different parts of the book as independent as possible, with only one exception: The functional calculus for self-adjoint operators developed in Sect. 5.3 is used as an essential tool throughout.

The book contains a number of important subjects (Krein’s spectral shift, boundary triplets, the theory of positive self-adjoint extensions, and others) and technical topics (the tensor product of unbounded operators, commutativity of self-adjoint operators, the bounded transform, Aronzajn–Donoghue theory) which are rarely if ever presented in text books. It is particularly hoped that the material presented will be found to be useful for graduate students and young researchers in mathematics and mathematical physics.

Advanced courses on unbounded self-adjoint operators can be built on this book.

One should probably start with the general theory of closed operators by presenting the core material of Sects. 1.1, 1.2, 1.3, 2.1, 2.2, 2.3, 3.1, and 3.2. This could be followed by spectral integrals and the spectral theorem for unbounded self-adjoint operators based on selected material from Sects. 4.1, 4.2, 4.3 and 5.2, 5.3, avoiding technical subtleties. There are many possibilities to continue. One could choose rel- atively bounded perturbations and Schrödinger operators (Chap. 8), or positive form and their applications (Chap. 10), or unitary groups (Sect. 6.1), or von Neumann’s

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extension theory (Sects. 13.1, 13.2), or linear relations (Sect. 14.1) and boundary triplets (Chap. 14). A large number of special topics treated in the book could be used as a part of an advanced course or a seminar.

The prerequisites for this book are the basics in functional analysis and of the theory of bounded Hilbert space operators as covered by a standard one semester course on functional analysis, together with a good working knowledge of measure theory. The applications on differential operators require some knowledge of or- dinary and partial differential equations and of Sobolev spaces. In Chaps. 9 and 16 a few selected results from complex analysis are also needed. For the convenience of the reader, we have added six appendices; on bounded operators and classes of compact operators, on measure theory, on the Fourier transform, on Sobolev spaces, on absolutely continuous functions, and on Stieltjes transforms and Nevanlinna functions. These collect a number of special results that are used at various places in the text. For the results, here we have provided either precise references to standard works or complete proofs.

A few general notations that are repeatedly used are listed after the table of contents. A more detailed symbol index can be found at the end of the book. Occasion- ally, I have used either simplified or overlapping notations, and while this might at first sight seem careless, the meaning will be always clear from the context. Thus the symbolx denotes a Hilbert space vector in one section and a real variable or even the functionf (x)=xin others.

A special feature of the book is the inclusion of numerous examples which are developed in detail and which are accompanied by exercises at the ends of the chapters. A number of simple standard examples (for instance, multiplication operators or differential operators−i_dx^d or −_dx^d²2 on intervals with different boundary conditions) are guiding examples and appear repeatedly throughout. They are used to illustrate various methods of constructing self-adjoint operators, as well as new notions even within the advanced chapters. The reader might also consider some examples as exercises with solutions and try to prove the statements therein first by himself, comparing the results with the given proofs. Often statements of exercises provide additional information concerning the theory. The reader who is interested in acquiring an ability to work with unbounded operators is of course encouraged to work through as many of the examples and exercises as possible. I have marked the somewhat more difficult exercises with a star. All stated exercises (with the possible exception of a few starred problems) are really solvable by students, as can be attested to by my experience in teaching this material. The hints added to exercises always contain key tricks or steps for the solutions.

In the course of teaching this subject and writing this book, I have benefited from many excellent sources. I should mention the volumes by Reed and Simon [RS1, RS2, RS4], Kato’s monograph [Ka], and the texts (in alphabetic order) [AG, BSU, BEH, BS, D1, D2, DS, EE, RN, Yf, We]. I have made no attempt to give precise credit for a result, an idea or a proof, though the few names of standard theorems are stated in the body of the text. The notes at the end of each part contain some (certainly incomplete) information about a few sources, important papers and monographs in this area and hints for additional reading. Also I have listed a number

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Preface and Overview xi of key pioneering papers. I felt it might be of interest for the reader to look at some of these papers and to observe, for instance, that H. Weyl’s work around 1909–

1911 contains fundamental and deep results about Hilbert space operators, while the corresponding general theory was only developed some 20 years later!

I am deeply indebted to Mr. René Gebhardt who read large parts of this book carefully and made many valuable suggestions. Finally, I would like to thank D. Dubray and K. Zimmermann for reading parts of the manuscript and Prof. M.A.

Dritschel and Dr. Y. Savchuk for their help in preparing this book.

Konrad Schmüdgen Leipzig

December 2011

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General Notation

N0: set of nonnegative integers, N: set of positive integers, Z: set of integers,

R: set of real numbers, C: set of complex numbers,

T: set of complex numbers of modulus one, i: complex unit,

χ_M: characteristic function of a setM.

Forα=(α₁, . . . , α_d)∈N^d₀,k=1, . . . , d, andx=(x₁, . . . , x_d)∈R^d, we set x^α:=x₁^α¹· · ·x_d^α^d, |α| :=α₁+ · · · +α_d,

∂_k:= ∂

∂xk

, D_k:= −i ∂

∂xk

,

∂^α:=∂₁^α¹· · ·∂_d^α^d = ∂^α¹

∂x₁^α¹ · · · ∂^α^d

∂x_d^α^d, D^α:=D₁^α¹· · ·D^α_d^d =(−i)^|^α^| ∂^α¹

∂x₁^α¹ · · · ∂^α^d

∂x_d^α^d with the convention that terms withα_j=0 are set equal to one.

Sequences are written by round brackets such as(x_n)_∈Nor(x_n), while sets are denoted by braces such as{x_i:i∈I}. Pairs of elements are written as(x, y).

For an open subsetΩofR^d,

Cⁿ(Ω) is the set ofntimes continuously differentiable complex functions onΩ, C₀^∞(Ω) is the set of functions inC^∞(Ω)whose support is a compact subset ofΩ, Cⁿ(Ω) is the set of functionsf ∈Cⁿ(Ω)for which all functionsD^αf,|α| ≤n,

admit continuous extensions to the closureΩof the setΩinR^d, L^p(Ω) is theL^p-space with respect to the Lebesgue measure onΩ.

We writeL²(a, b)forL²((a, b))andCⁿ(a, b)forCⁿ((a, b)).

“a.e.” means “almost everywhere with respect to the Lebesgue measure.”

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The symbolHrefers to a Hilbert space with scalar product·,·and norm · . Scalar products are always denoted by angle brackets·,·. Occasionally, indices or subscripts such as·,·j or·,·_H²_(Ω)are added.

The symbol⊕stands for the orthogonal sum of Hilbert spaces, while ˙+means the direct sum of vector spaces. By a projection we mean an orthogonal projection.

σ (T ): spectrum ofT, ρ(T ): resolvent set ofT, π(T ): regularity domain ofT, D(T ): domain ofT,

N(T ): null space ofT, R(T ): range ofT,

Rλ(T ): resolvent(T−λI )⁻¹ofT atλ, ET: spectral measure ofT.

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Part I

Basics of Closed Operators

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Chapter 1 Closed and Adjoint Operators

Closed operators and closable operators are important classes of unbounded linear operators which are large enough to cover all interesting operators occurring in applications. In this chapter and the next, we develop basic concepts and results about general closed operators on Hilbert space. We define and study closed operators, closable operators, closures, and cores in Sect.1.1and adjoint operators in Sect.1.2, while in Sect.1.3these concepts are discussed for differentiation operators on intervals and for linear partial differential operators. Section1.4deals with invariant subspaces and reducing subspaces of linear operators.

1.1 Closed and Closable Operators and Their Graphs 1.1.1 General Notions on Linear Operators

Definition 1.1 A linear operator from a Hilbert spaceH1into a Hilbert spaceH2is a linear mappingT of a linear subspace ofH1, called the domain ofT and denoted byD(T ), intoH2.

It should be emphasized that the domain is crucial for an unbounded operator.

The same formal expression considered on different domains (for instance, by vary- ing boundary conditions for the same differential expression) may lead to operators with completely different properties. We will see this by the examples in Sect.1.3.

ThatT is a linear mapping means that T (αx+βy)=αT (x)+βT (y)for all x, y∈D(T )andα, β∈C. The linear subspace

R(T ):=T D(T )

=

T (x):x∈D(T ) ofH2is called the range or the image ofT, and the linear subspace

N(T )=

x∈D(T ):T (x)=0 K. Schmüdgen, Unbounded Self-adjoint Operators on Hilbert Space, Graduate Texts in Mathematics 265,

DOI10.1007/978-94-007-4753-1_1, © Springer Science+Business Media Dordrecht 2012 3

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ofH1is the null space or the kernel ofT. We also writeT xforT (x)if no confusion can arise.

By the restriction ofT to a linear subspace D0 of D(T ) we mean the linear operatorTD0with domainD0acting by(TD0)(x)=T (x)forx∈D0.

LetS andT be two linear operators fromH1into H2. By definition we have S=T if and only ifD(S)=D(T )andS(x)=T (x)for allx∈D(S)=D(T ). We shall say thatT is an extension ofSor thatSis a restriction ofT and writeS⊆T, or equivalentlyT ⊇S, whenD(S)⊆D(T )andS(x)=T (x)for allx∈D(S). That is, we haveS⊆T if and only ifS=TD(S).

The complex multipleαT forα∈C,α=0, and the sum S+T are the linear operators fromH1intoH2defined by

D(αT )=D(T ), (αT )(x)=αT (x) forx∈D(αT ), α=0,

D(S+T )=D(S)∩D(T ), (S+T )(x)=S(x)+T (x) forx∈D(S+T ).

Further, we define the multipleαT forα=0 to be the null operator 0 fromH1

intoH2; it has the domainD(0)=H1and acts by 0(x)=0 forx∈H1.

IfR is a linear operator fromH2into a Hilbert spaceH3, then the productRT is the linear operator fromH1intoH3given by

D(RT )=

x∈D(T ):T x∈D(R)

, (RT )(x)=R(T x) forx∈D(RT ).

It is easily checked that the sum and product of linear operators are associative and that the two distributivity laws

(S+T )Q=SQ+T Q and R(S+T )⊇RS+RT (1.1) hold, whereQis a linear operator from a Hilbert spaceH0intoH1; see Exercises 1 and 2.

IfN(T )= {0}, then the inverse operator T⁻¹ is the linear operator fromH2

intoH1defined byD(T⁻¹)=R(T )andT⁻¹(T (x))=xforx∈D(T ). In this case we say thatT is invertible. Note thatR(T⁻¹)=D(T ). Clearly, ifN(R)= {0}and N(T )= {0}, thenN(RT )= {0}and(RT )⁻¹=T⁻¹R⁻¹.

Definition 1.2 The graph of a linear operatorT fromH1intoH2is the set G(T )=

(x, T x):x∈D(T ) .

The graphG(T )is a linear subspace of the Hilbert spaceH1⊕H2 which contains the full information about the operatorT. Obviously, the relationS⊆T is equivalent to the inclusionG(S)⊆G(T ). The next lemma contains an internal char- acterization of graphs. We omit the simple proof.

Lemma 1.1 A linear subspaceEofH1⊕H2is the graph of a linear operatorT fromH1intoH2if and only if(0, y)∈Efory∈H2implies thaty=0.

The operator T is then uniquely determined by E; it acts by T x =y for (x, y)∈E, and its domain is D(T )= {x ∈H1: There existsy ∈H2such that (x, y)∈E}.

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1.1 Closed and Closable Operators and Their Graphs 5 Often we are dealing with a linear operator that maps a Hilbert space Hinto itself. We usually express this by saying that we have an operator onH. That is, whenever we speak about an operatorT on a Hilbert spaceH, we mean a linear mappingT of a linear subspaceD(T )ofHintoH.

LetHbe a Hilbert space. The identity map ofHis denoted byI_Hor byI if no confusion is possible. Occasionally, we writeλinstead ofλ·I forλ∈C.

LetT be a linear operator onH. Then the powers ofT are defined inductively byTⁿ:=T (Tⁿ⁻¹)for n∈NandT⁰:=I. If p(x)=α_nxⁿ+ · · · +α₁x+α₀ is a complex polynomial,p(T )is defined by p(T )=α_nTⁿ+ · · · +α1T +α0I. If p1andp2are complex polynomials, thenp1(T )p2(T )=(p1p2)(T ). In particular, Tⁿ⁺^k=TⁿT^k=T^kTⁿfork, n∈NandD(p(T ))=D(Tⁿ)ifphas degreen.

IfT is an arbitrary linear operator onH, we have the polarization identity 4T x, y =

T (x+y), x+y

−

T (x−y), x−y +i

T (x+iy), x+iy

−i

T (x−iy), x−iy

(1.2) forx, y∈D(T ). It is proved by computing the right-hand side of (1.2).

Identity (1.2) is very useful and is often applied in this text. An immediate consequence of this identity is the following:

Lemma 1.2 LetT be a linear operator on Hsuch that D(T ) is dense inH. If T x, x =0 for allx∈D(T ), thenT x=0 for allx∈D(T ).

The following simple fact will be used several times in this book.

Lemma 1.3 LetSandT be linear operators such thatS⊆T. IfSis surjective and T is injective, thenS=T.

Proof Letx∈D(T ). SinceSis surjective, there is ay∈D(S)such thatT x=Sy.

FromS⊆T we getT x =T y, sox=y, becauseT is injective. Hence, we have x=y∈D(S). Thus,D(T )⊆D(S), whenceS=T. Except for the polarization identity and Lemma1.2, all preceding notions and facts required only the vector space structure. In the next subsection we essentially use the norm and completeness of the underlying Hilbert spaces.

1.1.2 Closed and Closable Operators

LetT be a linear operator from a Hilbert space H1 into a Hilbert spaceH2. Let

·,· j and·j denote the scalar product and norm ofHj. It is easily seen that x, y _T = x, y ₁+ T x, T y ₂, x, y∈D(T ), (1.3) defines a scalar product on the domainD(T ). The corresponding norm

xT =

x²₁+ T x²₂1/2

, x∈D(T ), (1.4)

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is called the graph norm of the operatorT. It is equivalent to the norm x_T := x1+ T x2, x∈D(T ),

onD(T ). Occasionally, the norm · _T is called the graph norm ofT as well.

Definition 1.3 An operatorT is called closed if its graphG(T )is a closed subset of the Hilbert spaceH1⊕H2, andT is called closable (or preclosed) if there exists a closed linear operatorSfromH1intoH2such thatT ⊆S.

The next propositions contain some slight reformulations of these definitions.

Proposition 1.4 The following statements are equivalent:

(i) T is closed.

(ii) If(xn)_n∈Nis a sequence of vectorsxn∈D(T )such that limnxn=xinH1and limnT (xn)=yinH2, thenx∈D(T )andT x=y.

(iii) (D(T ), · T)is complete, or equivalently,(D(T ),·,· T)is a Hilbert space.

Proof (i) is equivalent to (ii), because (ii) is only a reformulation of the closedness of the graphG(T )inH1⊕H2.

(i)↔(iii): By the definition (1.4) of the graph norm, the mapx→(x, T x)of (D(T ),·T)onto the graphG(T )equipped with the norm of the direct sum Hilbert spaceH1⊕H2 is isometric. Therefore,(D(T ), · T)is complete if and only if G(T )is complete, or equivalently, ifG(T )is closed inH1⊕H2. Proposition 1.5 The following are equivalent:

(i) T is closable.

(ii) If(x_n)_n_∈Nis a sequence of vectorsx_n∈D(T )such that limnx_n=0 inH1and limnT (x_n)=yinH2, theny=0.

(iii) The closure of the graphG(T )is the graph of a linear operator.

Proof (i)→(iii): LetSbe a closed operator such thatT ⊆S. ThenG(T )⊆G(S), and so G(T )⊆G(S). From Lemma 1.1it follows that G(T ) is the graph of an operator.

(iii)→(i): If G(T )=G(S)for some operator S, thenT ⊆S, and Sis closed, becauseG(S)is closed.

(ii)↔(iii): Condition (ii) means that(0, y)∈G(T )implies thaty=0. Hence,

(ii) and (iii) are equivalent by Lemma1.1.

Suppose thatT is closable. LetT denote the closed linear operator such that G(T )=G(T )

by Proposition 1.5(iii). By this definition, D(T ) is the set of vectors x ∈H1

for which there exists a sequence (x_n)_n_∈N from D(T ) such that x =limx_n in H1 and (T (x_n))_n_∈N converges in H2. For such a sequence (x_n), we then set

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1.1 Closed and Closable Operators and Their Graphs 7 T (x):=limnT (x_n). SinceG(T )is the graph of an operator, this definition ofT (x) is independent of the sequence(x_n). Clearly,T is the smallest (with respect to the operator inclusion⊆) closed extension of the closable operatorT.

Definition 1.4 The operatorT is called the closure of the closable operatorT. Let us briefly discuss these concepts for continuous operators. Recall that a linear operatorT is continuous (that is, if limnx_n=xinH1forx_n, x∈D(T ),n∈N, then limnT (x_n)=T (x)inH2) if and only ifT is bounded (that is, there is a constant c >0 such thatT (x)2≤cx1for allx∈D(T )).

If an operatorT is continuous, then it is clearly closable by Proposition1.5(ii). In fact, closability can be considered as a weakening of continuity for linear operators.

Let(xn)n∈Nbe a sequence fromD(T )such that limnxn=0 inH1. If the operatorT is continuous, then limnT (xn)=0 in H2. ForT being closable, we require only that if (!) the sequence(T (x_n))_n_∈Nconverges inH2, then it has to converge to the

“correct limit”, that is, limnT (x_n)=0.

IfT is bounded, the graph norm · T onD(T )is obviously equivalent to the Hilbert space norm · 1. Therefore, a bounded linear operatorT is closed if and only if its domainD(T )is closed inH1. Conversely, ifT is a closed linear operator whose domainD(T )is closed inH1, then the closed graph theorem implies that T is bounded. In particular, it follows that a closed linear operator which is not bounded cannot be defined on the whole Hilbert space.

Another useful notion is that of a core of an operator.

Definition 1.5 A linear subspaceDofD(T )is called a core forT ifDis dense in (D(T ),·T), that is, for eachx∈D(T ), there exists a sequence(xn)n∈Nof vectors xn∈Dsuch thatx=limnxninH1andT x=limnT (xn)inH2.

IfT is closed, a linear subspaceDof D(T )is a core forT if and only ifT is the closure of its restrictionTD. That is, a closed operator can be restored from its restriction to any core. The advantage of a core is that closed operators are often easier to handle on appropriate cores rather than on full domains.

Example 1.1 (Nonclosable operators) Let D be a linear subspace of a Hilbert spaceH, and lete=0 be a vector ofH. LetF be a linear functional onDwhich is not continuous in the Hilbert space norm. Define the operatorT byD(T )=Dand T (x)=F (x)eforx∈D.

Statement T is not closable.

Proof Since F is not continuous, there exists a sequence (xn)n∈N from D such that limnx_n=0 in H and (F (x_n)) does not converge to zero. By passing to a subsequence if necessary we can assume that there is a constantc >0 such that

|F (x_n)| ≥c for all n∈N. Putting x_n =F (x_n)⁻¹x_n, we have limnx_n =0 and T (x_n)=F (x_n)e=e=0. Hence,T is not closable by Proposition1.5(ii).

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The preceding proof has also shown that(0, e)∈G(T ), soG(T )is not the graph of a linear operator.

Explicit examples of discontinuous linear functionals are easily obtained as follows: If D is the linear span of an orthonormal sequence (e_n)_n_∈N of a Hilbert space, defineF on DbyF (e_n)=1, n∈N. IfH=L²(R)andD=C₀^∞(R), de-

fineF (f )=f (0)forf ∈D. ◦

1.2 Adjoint Operators

In this section the scalar products of the underlying Hilbert spaces are essentially used to define adjoints of densely defined linear operators.

Let(H1,·,· 1)and(H2,·,· 2) be Hilbert spaces. LetT be a linear operator fromH1intoH2such that the domainD(T )is dense inH1. Set

D T^∗

=

y∈H2:There existsu∈H1such thatT x, y ₂= x, u₁ forx∈D(T )

.

By Riesz’ theorem, a vector y ∈H2 belongs to D(T^∗) if and only if the map x→ T x, y ₂is a continuous linear functional on(D(T ), · 1), or equivalently, there is a constantcy>0 such that|T x, y 2| ≤cyx1for allx∈D(T ). An ex- plicit description of the setD(T^∗)is in general a very difficult matter.

SinceD(T )is dense inH1, the vectoru∈H1satisfyingT x, y 2= x, u1for allx∈D(T )is uniquely determined byy. Therefore, settingT^∗y=u, we obtain a well-defined mappingT^∗fromH2intoH1. It is easily seen thatT^∗is linear.

Definition 1.6 The linear operatorT^∗is called the adjoint operator ofT. By the preceding definition we have

T x, y 2= x, T^∗y

1 for allx∈D(T ), y∈D T^∗

. (1.5)

LetT be a densely defined linear operator onH. ThenT is called symmetric if T ⊆T^∗. Further, we say thatT is self-adjoint ifT =T^∗and thatT is essentially self-adjoint if its closureT is self-adjoint. These are fundamental notions studied extensively in this book.

The domainD(T^∗)of T^∗may be not dense inH2as the next example shows.

There are even operatorsT such thatD(T^∗)consists of the null vector only.

Example 1.2 (Example1.1continued) Suppose thatD(T )is dense inH. Since the functionalF is discontinuous, the mapx→ T x, y =F (x)e, y is continuous if and only ify⊥e. Hence,D(T^∗)=e^⊥inHandT^∗y=0 fory∈D(T^∗). ◦ Example 1.3 (Multiplication operators by continuous functions) LetJ be an in- terval. For a continuous functionϕonJ, we define the multiplication operatorMϕ

on the Hilbert spaceL²(J)by

(M_ϕf )(x)=ϕ(x)f (x) forf ∈D(M_ϕ):=

f∈L²(J):ϕ·f ∈L²(J) .

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1.2 Adjoint Operators 9 SinceD(M_ϕ)contains all continuous functions with compact support,D(M_ϕ)is dense, so the adjoint operator(M_ϕ)^∗exists. We will prove the following:

Statement (Mϕ)^∗=M_ϕ. Proof From the relation

Mϕf, g =

Jϕf g dx=

Jf ϕg dx= f, Mϕg forf, g∈D(M_ϕ)=D(M_ϕ)we conclude thatM_ϕ⊆(M_ϕ)^∗.

To prove the converse inclusion, letg∈D((Mϕ)^∗)and seth=(Mϕ)^∗g. LetχK

be the characteristic function of a compact subsetKofJ. Forf ∈D(Mϕ), we have f·χ_K∈D(M_ϕ)andM_ϕ(f χ_K), g = ϕf χ_K, g = f χ_K, h , so that

Jf χ_K(ϕg−h) dx=0.

SinceD(M_ϕ)is dense, the element χ_K(ϕg−h) of L²(J)must be zero, so that ϕg=honKand hence on the whole intervalJ. That is, we haveg∈D(M_ϕ)and h=(Mϕ)^∗g=Mϕg. This completes the proof of the equality(Mϕ)^∗=Mϕ. The special case whereϕ(x)=x andJ =Ris of particular importance. The operatorQ:=M_ϕ =M_x is then the position operator of quantum mechanics. By the preceding statement we haveQ=Q^∗, soQis a self-adjoint operator onL²(R).

Multiplication operators by measurable functions on general measure spaces will

be studied in Sect. 3.4. ◦

We now begin to develop basic properties of adjoint operators.

Proposition 1.6 LetSandT be linear operators fromH1intoH2such thatD(T ) is dense inH1. Then:

(i) T^∗is a closed linear operator fromH2intoH1. (ii) R(T )^⊥=N(T^∗).

(iii) IfD(T^∗)is dense inH2, thenT ⊆T^∗∗, whereT^∗∗:=(T^∗)^∗. (iv) IfT ⊆S, thenS^∗⊆T^∗.

(v) (λT )^∗=λT^∗forλ∈C.

(vi) IfD(T+S)is dense inH1, then(T+S)^∗⊇T^∗+S^∗. (vii) IfSis bounded andD(S)=H1, then(T+S)^∗=T^∗+S^∗.

Proof (i): Let(y_n)_n_∈Nbe a sequence fromD(T^∗)such that limny_n=y inH2and limnT^∗y_n=vinH1. Forx∈D(T ), we then have

T x, y_n =

x, T^∗y_n

→ T x, y = x, v , so thaty∈D(T^∗)andv=T^∗y. This proves thatT^∗is closed.

(ii): By (1.5) it is obvious that y∈N(T^∗)if and only if T x, y ₂=0 for all x∈D(T ), or equivalently,y∈R(T )^⊥. That is,R(T )^⊥=N(T^∗).

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(iii)–(vi) follow from the definition of the adjoint operator. We omit the details.

(vii): By (vi) it suffices to prove thatD((T+S)^∗)⊆D(T^∗+S^∗). SinceD(S)= H1 by assumption andD(S^∗)=H1 becauseS is bounded, we haveD(T +S)= D(T )andD(T^∗+S^∗)=D(T^∗). Lety∈D((T+S)^∗). Forx∈D(T+S),

T x, y =

(T+S)x, y

− Sx, y =

x, (T+S)^∗y−S^∗y .

This implies thaty∈D(T^∗)=D(T^∗+S^∗).

Let T be a densely defined closable linear operator from H1 into H2. Then T =(T^∗)^∗ as shown in Theorem 1.8(ii) below. Taking this for granted, Proposi- tion1.6(ii), applied toT andT^∗, yields

N T^∗

=R(T )^⊥, N(T )=R T^∗_⊥

, H2=N

T^∗

⊕R(T ), H1=N(T )⊕R T^∗

.

(1.6) Note that the rangesR(T )andR(T^∗)are not closed in general. However, the closed range theorem (see, e.g., [K2, IV, Theorem 5.13]) states that for a closed linear operatorT, the rangeR(T )is closed inH2if and only ifR(T^∗)is closed inH1.

ReplacingT byT −λI in (1.6), one obtains the useful decomposition H=N

T^∗−λI

⊕R(T −λI )=N(T −λI )⊕R

T^∗−λI

(1.7) for any densely defined closable operatorT on a Hilbert spaceHandλ∈C.

Assertions (vi) and (vii) of Proposition1.6have counterparts for the product.

Proposition 1.7 LetT :H1→H2andS:H2→H3be linear operators such that D(ST )is dense inH1.

(i) IfD(S)is dense inH2, then(ST )^∗⊇T^∗S^∗.

(ii) IfSis bounded andD(S)=H2, then(ST )^∗=T^∗S^∗.

Proof (i): Note thatD(T )⊇D(ST ); hence, D(T ) is dense inH2. Suppose that y∈D(T^∗S^∗). Letx∈D(ST ). Then we haveT x∈D(S),y∈D(S^∗), and

ST x, y =

T x, S^∗y

=

x, T^∗S^∗y . Therefore,y∈D((ST )^∗)and(ST )^∗y=T^∗S^∗y.

(ii): Because of (i), it is sufficient to show thatD((ST )^∗)⊆D(T^∗S^∗). Suppose thaty ∈D((ST )^∗). Letx∈D(T ). SinceS is bounded and D(S)=H1, we have x∈D(ST )andy∈D(S^∗)=H2, so that

T x, S^∗y

= ST x, y =

x, (ST )^∗y .

Consequently,S^∗y∈D(T^∗), and soy∈D(T^∗S^∗).

A number of important and useful properties of closable and adjoint operators are derived by the so-called graph method. They are collected in the following theorem.

Clearly, assertions (v) and (vi) are still valid ifD(T )is not dense inH1.

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1.2 Adjoint Operators 11 Theorem 1.8 LetT be a densely defined linear operator fromH1intoH2.

(i) T is closable if and only ifD(T^∗)is dense inH2.

(ii) IfT is closable, then(T )^∗=T^∗, and settingT^∗∗:=(T^∗)^∗, we have T =T^∗∗.

(iii) T is closed if and only ifT =T^∗∗.

(iv) Suppose thatN(T )= {0}andR(T )is dense inH2. ThenT^∗is invertible and T^∗₋1

= T⁻¹_∗

.

(v) Suppose thatT is closable and N(T )= {0}. Then the inverseT⁻¹ of T is closable if and only ifN(T )= {0}. If this holds, then

(T )⁻¹= T⁻¹

.

(vi) IfT is invertible, thenT is closed if and only ifT⁻¹is closed.

Before we turn to the proof we state an important consequence separately as the following:

Corollary 1.9 IfT is a self-adjoint operator such that N(T )= {0}, then T⁻¹is also a self-adjoint operator.

Proof SinceT =T^∗, we haveR(T )^⊥=N(T )= {0}. Hence,R(T )is dense, and

the assertion follows from Theorem1.8(iv).

The main technical tool for the graph method are the two unitary operatorsU,V ofH1⊕H2ontoH2⊕H1defined by

U (x, y)=(y, x) and V (x, y)=(−y, x) forx∈H1, y∈H2. (1.8) Clearly,V⁻¹(y, x)=(x,−y)fory∈H2andx∈H1, andU⁻¹V =V⁻¹U, since

U⁻¹V (x, y)=U⁻¹(−y, x)=(x,−y)=V⁻¹(y, x)=V⁻¹U (x, y).

LetT be an invertible operator fromH1intoH2. Ifx∈D(T )andy=T x, then U (x, T x)=(y, T⁻¹y). Therefore,Umaps the graph ofT onto the graph ofT⁻¹, that is,

U G(T )

=G T⁻¹

. (1.9)

The next lemma describes the graph ofT^∗ by means of the unitaryV. In particular, it implies thatG(T^∗)is closed. This gives a second proof of the fact (see Proposition1.6(i)) that the operatorT^∗is closed.

Lemma 1.10 For any densely defined linear operatorT ofH1intoH2, we have G

T^∗

=V G(T )_⊥

=V G(T )^⊥

.

Graduate Texts in Mathematics 265

Graduate Texts in Mathematics 265

Graduate Texts in Mathematics

Konrad Schmüdgen

Unbounded

Self-adjoint Operators

on Hilbert Space

To ELISA

Preface and Overview

Contents

General Notation

Part I

Basics of Closed Operators

Chapter 1

Closed and Adjoint Operators

1.1 Closed and Closable Operators and Their Graphs 1.1.1 General Notions on Linear Operators

1.1.2 Closed and Closable Operators

1.2 Adjoint Operators