11. Unbounded Operators and Relations Many Important Operators on Hilbert Spaces Are Not Bounded
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1 Bounded and Unbounded Operators
1 1 Bounded and unbounded operators 1. Let X, Y be Banach spaces and D 2 X a linear space, not necessarily closed. 2. A linear operator is any linear map T : D ! Y . 3. D is the domain of T , sometimes written Dom (T ), or D (T ). 4. T (D) is the Range of T , Ran(T ). 5. The graph of T is Γ(T ) = f(x; T x)jx 2 D (T )g 6. The kernel of T is Ker(T ) = fx 2 D (T ): T x = 0g 1.1 Operations 1. aT1 + bT2 is defined on D (T1) \D (T2). 2. if T1 : D (T1) ⊂ X ! Y and T2 : D (T2) ⊂ Y ! Z then T2T1 : fx 2 D (T1): T1(x) 2 D (T2). In particular, inductively, D (T n) = fx 2 D (T n−1): T (x) 2 D (T )g. The domain may become trivial. 3. Inverse. The inverse is defined if Ker(T ) = f0g. This implies T is bijective. Then T −1 : Ran(T ) !D (T ) is defined as the usual function inverse, and is clearly linear. @ is not invertible on C1[0; 1]: Ker@ = C. 4. Closable operators. It is natural to extend functions by continuity, when possible. If xn ! x and T xn ! y we want to see whether we can define T x = y. Clearly, we must have xn ! 0 and T xn ! y ) y = 0; (1) since T (0) = 0 = y. Conversely, (1) implies the extension T x := y when- ever xn ! x and T xn ! y is consistent and defines a linear operator. -
Adjoint of Unbounded Operators on Banach Spaces
November 5, 2013 ADJOINT OF UNBOUNDED OPERATORS ON BANACH SPACES M.T. NAIR Banach spaces considered below are over the field K which is either R or C. Let X be a Banach space. following Kato [2], X∗ denotes the linear space of all continuous conjugate linear functionals on X. We shall denote hf; xi := f(x); x 2 X; f 2 X∗: On X∗, f 7! kfk := sup jhf; xij kxk=1 defines a norm on X∗. Definition 1. The space X∗ is called the adjoint space of X. Note that if K = R, then X∗ coincides with the dual space X0. It can be shown, analogues to the case of X0, that X∗ is a Banach space. Let X and Y be Banach spaces, and A : D(A) ⊆ X ! Y be a densely defined linear operator. Now, we st out to define adjoint of A as in Kato [2]. Theorem 2. There exists a linear operator A∗ : D(A∗) ⊆ Y ∗ ! X∗ such that hf; Axi = hA∗f; xi 8 x 2 D(A); f 2 D(A∗) and for any other linear operator B : D(B) ⊆ Y ∗ ! X∗ satisfying hf; Axi = hBf; xi 8 x 2 D(A); f 2 D(B); D(B) ⊆ D(A∗) and B is a restriction of A∗. Proof. Suppose D(A) is dense in X. Let S := ff 2 Y ∗ : x 7! hf; Axi continuous on D(A)g: For f 2 S, define gf : D(A) ! K by (gf )(x) = hf; Axi 8 x 2 D(A): Since D(A) is dense in X, gf has a unique continuous conjugate linear extension to all ∗ of X, preserving the norm. -
On the Origin and Early History of Functional Analysis
U.U.D.M. Project Report 2008:1 On the origin and early history of functional analysis Jens Lindström Examensarbete i matematik, 30 hp Handledare och examinator: Sten Kaijser Januari 2008 Department of Mathematics Uppsala University Abstract In this report we will study the origins and history of functional analysis up until 1918. We begin by studying ordinary and partial differential equations in the 18th and 19th century to see why there was a need to develop the concepts of functions and limits. We will see how a general theory of infinite systems of equations and determinants by Helge von Koch were used in Ivar Fredholm’s 1900 paper on the integral equation b Z ϕ(s) = f(s) + λ K(s, t)f(t)dt (1) a which resulted in a vast study of integral equations. One of the most enthusiastic followers of Fredholm and integral equation theory was David Hilbert, and we will see how he further developed the theory of integral equations and spectral theory. The concept introduced by Fredholm to study sets of transformations, or operators, made Maurice Fr´echet realize that the focus should be shifted from particular objects to sets of objects and the algebraic properties of these sets. This led him to introduce abstract spaces and we will see how he introduced the axioms that defines them. Finally, we will investigate how the Lebesgue theory of integration were used by Frigyes Riesz who was able to connect all theory of Fredholm, Fr´echet and Lebesgue to form a general theory, and a new discipline of mathematics, now known as functional analysis. -
Proved for Real Hilbert Spaces. Time Derivatives of Observables and Applications
AN ABSTRACT OF THE THESIS OF BERNARD W. BANKSfor the degree DOCTOR OF PHILOSOPHY (Name) (Degree) in MATHEMATICS presented on (Major Department) (Date) Title: TIME DERIVATIVES OF OBSERVABLES AND APPLICATIONS Redacted for Privacy Abstract approved: Stuart Newberger LetA andH be self -adjoint operators on a Hilbert space. Conditions for the differentiability with respect totof -itH -itH <Ae cp e 9>are given, and under these conditionsit is shown that the derivative is<i[HA-AH]e-itHcp,e-itHyo>. These resultsare then used to prove Ehrenfest's theorem and to provide results on the behavior of the mean of position as a function of time. Finally, Stone's theorem on unitary groups is formulated and proved for real Hilbert spaces. Time Derivatives of Observables and Applications by Bernard W. Banks A THESIS submitted to Oregon State University in partial fulfillment of the requirements for the degree of Doctor of Philosophy June 1975 APPROVED: Redacted for Privacy Associate Professor of Mathematics in charge of major Redacted for Privacy Chai an of Department of Mathematics Redacted for Privacy Dean of Graduate School Date thesis is presented March 4, 1975 Typed by Clover Redfern for Bernard W. Banks ACKNOWLEDGMENTS I would like to take this opportunity to thank those people who have, in one way or another, contributed to these pages. My special thanks go to Dr. Stuart Newberger who, as my advisor, provided me with an inexhaustible supply of wise counsel. I am most greatful for the manner he brought to our many conversa- tions making them into a mutual exchange between two enthusiasta I must also thank my parents for their support during the earlier years of my education.Their contributions to these pages are not easily descerned, but they are there never the less. -
On Relaxation of Normality in the Fuglede-Putnam Theorem
proceedings of the american mathematical society Volume 77, Number 3, December 1979 ON RELAXATION OF NORMALITY IN THE FUGLEDE-PUTNAM THEOREM TAKAYUKIFURUTA1 Abstract. An operator means a bounded linear operator on a complex Hubert space. The familiar Fuglede-Putnam theorem asserts that if A and B are normal operators and if X is an operator such that AX = XB, then A*X = XB*. We shall relax the normality in the hypotheses on A and B. Theorem 1. If A and B* are subnormal and if X is an operator such that AX = XB, then A*X = XB*. Theorem 2. Suppose A, B, X are operators in the Hubert space H such that AX = XB. Assume also that X is an operator of Hilbert-Schmidt class. Then A*X = XB* under any one of the following hypotheses: (i) A is k-quasihyponormal and B* is invertible hyponormal, (ii) A is quasihyponormal and B* is invertible hyponormal, (iii) A is nilpotent and B* is invertible hyponormal. 1. In this paper an operator means a bounded linear operator on a complex Hilbert space. An operator T is called quasinormal if F commutes with T* T, subnormal if T has a normal extension and hyponormal if [ F*, T] > 0 where [S, T] = ST - TS. The inclusion relation of the classes of nonnormal opera- tors listed above is as follows: Normal § Quasinormal § Subnormal ^ Hyponormal; the above inclusions are all proper [3, Problem 160, p. 101]. The familiar Fuglede-Putnam theorem is as follows ([3, Problem 152], [4]). Theorem A. If A and B are normal operators and if X is an operator such that AX = XB, then A*X = XB*. -
The Notion of Observable and the Moment Problem for ∗-Algebras and Their GNS Representations
The notion of observable and the moment problem for ∗-algebras and their GNS representations Nicol`oDragoa, Valter Morettib Department of Mathematics, University of Trento, and INFN-TIFPA via Sommarive 14, I-38123 Povo (Trento), Italy. [email protected], [email protected] February, 17, 2020 Abstract We address some usually overlooked issues concerning the use of ∗-algebras in quantum theory and their physical interpretation. If A is a ∗-algebra describing a quantum system and ! : A ! C a state, we focus in particular on the interpretation of !(a) as expectation value for an algebraic observable a = a∗ 2 A, studying the problem of finding a probability n measure reproducing the moments f!(a )gn2N. This problem enjoys a close relation with the selfadjointness of the (in general only symmetric) operator π!(a) in the GNS representation of ! and thus it has important consequences for the interpretation of a as an observable. We n provide physical examples (also from QFT) where the moment problem for f!(a )gn2N does not admit a unique solution. To reduce this ambiguity, we consider the moment problem n ∗ (a) for the sequences f!b(a )gn2N, being b 2 A and !b(·) := !(b · b). Letting µ!b be a n solution of the moment problem for the sequence f!b(a )gn2N, we introduce a consistency (a) relation on the family fµ!b gb2A. We prove a 1-1 correspondence between consistent families (a) fµ!b gb2A and positive operator-valued measures (POVM) associated with the symmetric (a) operator π!(a). In particular there exists a unique consistent family of fµ!b gb2A if and only if π!(a) is maximally symmetric. -
Hilbert Space Quantum Mechanics
qmd113 Hilbert Space Quantum Mechanics Robert B. Griffiths Version of 27 August 2012 Contents 1 Vectors (Kets) 1 1.1 Diracnotation ................................... ......... 1 1.2 Qubitorspinhalf ................................. ......... 2 1.3 Intuitivepicture ................................ ........... 2 1.4 General d ............................................... 3 1.5 Ketsasphysicalproperties . ............. 4 2 Operators 5 2.1 Definition ....................................... ........ 5 2.2 Dyadsandcompleteness . ........... 5 2.3 Matrices........................................ ........ 6 2.4 Daggeroradjoint................................. .......... 7 2.5 Hermitianoperators .............................. ........... 7 2.6 Projectors...................................... ......... 8 2.7 Positiveoperators............................... ............ 9 2.8 Unitaryoperators................................ ........... 9 2.9 Normaloperators................................. .......... 10 2.10 Decompositionoftheidentity . ............... 10 References: CQT = Consistent Quantum Theory by Griffiths (Cambridge, 2002). See in particular Ch. 2; Ch. 3; Ch. 4 except for Sec. 4.3. 1 Vectors (Kets) 1.1 Dirac notation ⋆ In quantum mechanics the state of a physical system is represented by a vector in a Hilbert space: a complex vector space with an inner product. The term “Hilbert space” is often reserved for an infinite-dimensional inner product space having the property◦ that it is complete or closed. However, the term is often used nowadays, as in these notes, in a way that includes finite-dimensional spaces, which automatically satisfy the condition of completeness. ⋆ We will use Dirac notation in which the vectors in the space are denoted by v , called a ket, where v is some symbol which identifies the vector. | One could equally well use something like v or v. A multiple of a vector by a complex number c is written as c v —think of it as analogous to cv of cv. | ⋆ In Dirac notation the inner product of the vectors v with w is written v w . -
Multiple Operator Integrals: Development and Applications
Multiple Operator Integrals: Development and Applications by Anna Tomskova A thesis submitted for the degree of Doctor of Philosophy at the University of New South Wales. School of Mathematics and Statistics Faculty of Science May 2017 PLEASE TYPE THE UNIVERSITY OF NEW SOUTH WALES Thesis/Dissertation Sheet Surname or Family name: Tomskova First name: Anna Other name/s: Abbreviation for degree as given in the University calendar: PhD School: School of Mathematics and Statistics Faculty: Faculty of Science Title: Multiple Operator Integrals: Development and Applications Abstract 350 words maximum: (PLEASE TYPE) Double operator integrals, originally introduced by Y.L. Daletskii and S.G. Krein in 1956, have become an indispensable tool in perturbation and scattering theory. Such an operator integral is a special mapping defined on the space of all bounded linear operators on a Hilbert space or, when it makes sense, on some operator ideal. Throughout the last 60 years the double and multiple operator integration theory has been greatly expanded in different directions and several definitions of operator integrals have been introduced reflecting the nature of a particular problem under investigation. The present thesis develops multiple operator integration theory and demonstrates how this theory applies to solving of several deep problems in Noncommutative Analysis. The first part of the thesis considers double operator integrals. Here we present the key definitions and prove several important properties of this mapping. In addition, we give a solution of the Arazy conjecture, which was made by J. Arazy in 1982. In this part we also discuss the theory in the setting of Banach spaces and, as an application, we study the operator Lipschitz estimate problem in the space of all bounded linear operators on classical Lp-spaces of scalar sequences. -
Class Notes, Functional Analysis 7212
Class notes, Functional Analysis 7212 Ovidiu Costin Contents 1 Banach Algebras 2 1.1 The exponential map.....................................5 1.2 The index group of B = C(X) ...............................6 1.2.1 p1(X) .........................................7 1.3 Multiplicative functionals..................................7 1.3.1 Multiplicative functionals on C(X) .........................8 1.4 Spectrum of an element relative to a Banach algebra.................. 10 1.5 Examples............................................ 19 1.5.1 Trigonometric polynomials............................. 19 1.6 The Shilov boundary theorem................................ 21 1.7 Further examples....................................... 21 1.7.1 The convolution algebra `1(Z) ........................... 21 1.7.2 The return of Real Analysis: the case of L¥ ................... 23 2 Bounded operators on Hilbert spaces 24 2.1 Adjoints............................................ 24 2.2 Example: a space of “diagonal” operators......................... 30 2.3 The shift operator on `2(Z) ................................. 32 2.3.1 Example: the shift operators on H = `2(N) ................... 38 3 W∗-algebras and measurable functional calculus 41 3.1 The strong and weak topologies of operators....................... 42 4 Spectral theorems 46 4.1 Integration of normal operators............................... 51 4.2 Spectral projections...................................... 51 5 Bounded and unbounded operators 54 5.1 Operations.......................................... -
Riesz-Like Bases in Rigged Hilbert Spaces, in Preparation [14] Bonet, J., Fern´Andez, C., Galbis, A
RIESZ-LIKE BASES IN RIGGED HILBERT SPACES GIORGIA BELLOMONTE AND CAMILLO TRAPANI Abstract. The notions of Bessel sequence, Riesz-Fischer sequence and Riesz basis are generalized to a rigged Hilbert space D[t] ⊂H⊂D×[t×]. A Riesz- like basis, in particular, is obtained by considering a sequence {ξn}⊂D which is mapped by a one-to-one continuous operator T : D[t] → H[k · k] into an orthonormal basis of the central Hilbert space H of the triplet. The operator T is, in general, an unbounded operator in H. If T has a bounded inverse then the rigged Hilbert space is shown to be equivalent to a triplet of Hilbert spaces. 1. Introduction Riesz bases (i.e., sequences of elements ξn of a Hilbert space which are trans- formed into orthonormal bases by some bounded{ } operator withH bounded inverse) often appear as eigenvectors of nonself-adjoint operators. The simplest situation is the following one. Let H be a self-adjoint operator with discrete spectrum defined on a subset D(H) of the Hilbert space . Assume, to be more definite, that each H eigenvalue λn is simple. Then the corresponding eigenvectors en constitute an orthonormal basis of . If X is another operator similar to H,{ i.e.,} there exists a bounded operator T withH bounded inverse T −1 which intertwines X and H, in the sense that T : D(H) D(X) and XT ξ = T Hξ, for every ξ D(H), then, as it is → ∈ easily seen, the vectors ϕn with ϕn = Ten are eigenvectors of X and constitute a Riesz basis for . -
Some Elements of Functional Analysis
Some elements of functional analysis J´erˆome Le Rousseau January 8, 2019 Contents 1 Linear operators in Banach spaces 1 2 Continuous and bounded operators 2 3 Spectrum of a linear operator in a Banach space 3 4 Adjoint operator 4 5 Fredholm operators 4 5.1 Characterization of bounded Fredholm operators ............ 5 6 Linear operators in Hilbert spaces 10 Here, X and Y will denote Banach spaces with their norms denoted by k.kX , k.kY , or simply k.k when there is no ambiguity. 1 Linear operators in Banach spaces An operator A from X to Y is a linear map on its domain, a linear subspace of X, to Y . One denotes by D(A) the domain of this operator. An operator from X to Y is thus characterized by its domain and how it acts on this domain. Operators defined this way are usually referred to as unbounded operators. One writes (A, D(A)) to denote the operator along with its domain. The set of linear operators from X to Y is denoted by L (X,Y ). If D(A) is dense in X the operator is said to be densely defined. If D(A) = X one says that the operator A is on X to Y . The range of the operator is denoted by Ran(A), that is, Ran(A)= {Ax; x ∈ D(A)} ⊂ Y, 1 and its kernel, ker(A), is the set of all x ∈ D(A) such that Ax = 0. The graph of A, G(A), is given by G(A)= {(x, Ax); x ∈ D(A)} ⊂ X × Y. -
Dissipative Operators in a Banach Space
Pacific Journal of Mathematics DISSIPATIVE OPERATORS IN A BANACH SPACE GUNTER LUMER AND R. S. PHILLIPS Vol. 11, No. 2 December 1961 DISSIPATIVE OPERATORS IN A BANACH SPACE G. LUMER AND R. S. PHILLIPS 1. Introduction* The Hilbert space theory of dissipative operators1 was motivated by the Cauchy problem for systems of hyperbolic partial differential equations (see [5]), where a consideration of the energy of, say, an electromagnetic field leads to an L2 measure as the natural norm for the wave equation. However there are many interesting initial value problems in the theory of partial differential equations whose natural setting is not a Hilbert space, but rather a Banach space. Thus for the heat equation the natural measure is the supremum of the temperature whereas in the case of the diffusion equation the natural measure is the total mass given by an Lx norm. In the present paper a suitable extension of the theory of dissipative operators to arbitrary Banach spaces is initiated. An operator A with domain ®(A) contained in a Hilbert space H is called dissipative if (1.1) re(Ax, x) ^ 0 , x e ®(A) , and maximal dissipative if it is not the proper restriction of any other dissipative operator. As shown in [5] the maximal dissipative operators with dense domains precisely define the class of generators of strongly continuous semi-groups of contraction operators (i.e. bounded operators of norm =§ 1). In the case of the wave equation this furnishes us with a description of all solutions to the Cauchy problem for which the energy is nonincreasing in time.