Functional Analysis Exercise Class
|
|
- Evangeline Carter
- 5 years ago
- Views:
Transcription
1 Functional Analysis Exercise Class Week: January 18 Deadline to hand in the homework: your exercise class on week January 5 9. Exercises with solutions (1) a) Show that for every unitary operators U, V, the operators UV and U 1 are also unitaries. Conclude that the unitary operators on a Hilbert space form a group with respect to the operator multiplication and inverse. b) Show that every orthogonal projection is a partial isometry. c) Is it true that the product of two partial isometries/orthogonal projections is a partial isometry/orthogonal projection? a) Let U, V be unitaries, so that UU = U U = V V = V V = I. Then (UV ) (UV ) = V U UV = V V = I, (UV )(UV ) = UV V U = UU = I, so UV is a unitary. By definition, U 1 = U, and hence, (U 1 ) U 1 = (U ) U = UU = I, U 1 (U 1 ) = U U = I, so U 1 is a unitary. The conclusion about the group structure is obvious. b) Let P be a orthogonal projection; then (ker P ) = ran P, and we have P x = x for every x ran P. In particular, P is an isometry on (ker P ), and hence, by definition, it is a partial isometry. Alternatively, one can use the characterization that an operator A is a partial isometry if and only of A A is a orthogonal projection. Obviously, if P is a orthogonal projection then so is P P = P. c) No, in general the product of two orthogonal projections is not an orthogonal projection; by the above, this implies that the product of two partial isometries is not a partial isometry. As a concrete example, consider P := [ ], Q := 1 [ ], the projections onto the subspaces spanned by (1, 0) T and 1 (1, 1) T, respectively, in K. Then P Q = 1 [ ] 1 1, 0 0 which is easily seen not to be an orthogonal projection (for instance, it is not selfadjoint). 1
2 () Let H be a complex Hilbert-space. a) Show that if T B (H) is self-adjoint, and T 1, then U ± := T ± i(i T ) 1/ are unitaries. b) Prove that every T B (H) is the linear combination of at most 4 unitaries. (Hint: Use the following fact from the lecture: (I T ) 1/ commutes with every operator that commutes with I T.) a) As T 1, we have that T T 1, and hence, by the lecture, T I, i.e., I T 0. Hence, (I T ) 1/ is well-defined and commutes with every operator that commutes with I T (see the lecture). Since T commutes with I T, we see that (I T ) 1/ commutes with T. Thus, U ±U ± = (T ± i(i T ) 1/ )(T i(i T ) 1/ ) = T ± i(i T ) 1/ T T i(i T ) 1/ + (I T ) = T + (I T ) = I, and similarly, U ± U ± = I, i.e., U is a unitary. b) By a Lemma from the lecture we can decompose any T B (H) as T = A + ib with Hermitian operators A, B B (H). Now we apply (a) to the operators à = A and A B =. (If A = 0 then we define à := 0, and similarly for B.) This shows that B B U ± := à ± i(i à ) 1/ and V ± := B ± i(i B ) 1/ are unitaries. A direct computation shows that à = 1(U + + U ) and B = 1(V + + V ). Thus, we obtain the desired decomposition as T = A + ib = A (U + + U ) + i B (V + + V ). (3) A linear operator T on a Hilbert space H is called symmetric if x, T y = T x, y for all x, y H. Show that T is continuous and hence Hermitian. Using the closed graph theorem, it suffices to prove that for a sequence (x n ) n in H with x n 0 and T x n z, we have z = 0. But z, z = z, lim T x n = lim z, T x n = lim T z, x n = T z, lim x n = T z, 0 = 0. (4) (Multiplication operators) Let (, F, µ) be a σ-finite measure space, let H := L (, F, µ) be the Hilbert space of square-integrable functions with its standard inner product, i.e., f, g := f(x)ḡ(x), dµ(x).
3 Let φ L (, F, µ), and define the multiplication operator M φ f := φf, f L (, F, µ). Show that M φ is bounded, and the following hold: a) φ M φ is an algebra morphism such that M 1 = I b) (M φ ) = M φ c) M φ is normal. d) M φ1 = M φ φ 1 = φ µ-almost everywhere. e) M φ is self-adjoint φ(x) R for almost every x. f) M φ is unitary φ(x) = 1 for almost every x. g) M φ is positive φ(x) 0 for almost every x. h) M φ is an orthogonal projection φ is the indicator function of a measurable set. Remark: If you are not familiar with measure theory, then instead of the above abstract setting, take H to be the completion of C R ([0, 1]), take φ to be continuous, and skip (4)h) First, we show that φ L (, F, µ) guarantees that M φ is bounded. Indeed, M φ f = φ(x)f(x) dµ(x) φ f(x) dµ(x) = φ f, for any f L (, F, µ), showing that M φ φ. a) Let φ 1, φ L (, F, µ) and λ K. Then for every f L (, F, µ), M λφ1 +φ f (λm φ1 + M φ )f = (λφ 1 + φ )f (λφ 1 f + φ f) = 0, and hence M λφ1 +φ = λm φ1 + M φ. The proof of M φ1 φ = M φ1 M φ and M 1 = I goes the same way. b) For every f, g L (, F, µ), we have f, M φ g = M φ f, g = φ(x)f(x)g(x) dµ(x) = showing that (M φ ) = M φ. c) By the above, M φm φ = M φ M φ = M φ = M φ M φ = M φ M φ, and hence M φ is normal. f(x)φ(x)g(x) dµ(x) = f, M φ g, 3
4 d) By the linearity of φ M φ, it is enough to show that M φ = 0 = φ = 0 µ-almost everywhere. Let g(x) := φ(x)/ φ(x) when φ(x) 0, and g(x) := 0 otherwise. Then g is measurable. Since the measure space is σ-finite, there exist measurable sets n, n N, of finite measure such that n n =. For every n N, let f n := g1 n L (, F, µ), where 1 n is the indicator function of n. Assume that M φ = 0; then 0 = M φ f n = φ 1 n, n N, and hence φ(x) = 0 for almost every x n. Using that n n =, the assertion follows. e) We have M φ = (M φ ) = M φ φ = φ µ-almost everywhere. f) We have I = (M φ ) M φ = M φ φ = 1 µ-almost everywhere, and similary for I = M φ (M φ ). g) We have M φ f, f = φ(x) f(x) dµ(x), and hence if φ 0 µ-almost everywhere then M φ 0. Assume now that M φ 0. According to the lecture, this implies that M φ is self-adjoint, and hence, by the above, φ is real-valued µ-almost everywhere. Assume that := {x : φ(x) < 0} has non-zero measure. Then there exists a measurable subset A of finite measure (we are using the σ-finiteness here), and thus 1 A L (, F, µ). Moreover, M φ f, f = φ(x) f(x) dµ(x) = φ(x) dµ(x) < 0, contradicting the assumption that M φ 0. h) By definition, M φ is an orthogonal projection if and only if Mφ = M φ = Mφ, and by the above, this is equivalent to φ being real-valued, and φ(x) = φ(x) for µ-almost every x, which in turn is equivalent to φ taking only 0 or 1 value µ-almost everywhere, which is equivalent to φ being an indicator function of some set A. Since φ is measurable, A = {x : φ(x) = 1} is measurable, too. (5) (Lax-Milgram) Let B : H H K be a sesquilinear form that is bounded, i.e., B := sup{ B(x, y) : x = 1, y = 1} < +. Show the following: a) There exists a bounded linear operator B B(H) such that B(x, y) = x, By, x, y H, and B = B. b) If there exists a c > 0 such that B(x, x) c x, x H, then B is invertible, and B 1 c 1. Last year s lecture notes: page 61 at M5/Allgemeines/MA3001_014W/Hilbert4.pdf A 4
5 Homework with solutions (1) Show that the product of two orthogonal projections is an orthogonal projection if and only if they commute. Let P, Q be orthogonal projections. Then (P Q) = Q P = QP, and hence (P Q) = P Q QP = P Q, i.e., if P Q is an orthogonal projection then P and Q commute. To prove the opposite direction, assume that P and Q commute. Then, by the above, (P Q) = P Q, and (P Q) = P QP Q = P Q = P Q, and thus P Q is indeed an orthogonal projection. () Let A B(H). Show that the following are equivalent: (i) A is a self-adjoint unitary. (ii) There exists an orthogonal projection P such that A = P I. (iii) There exist orthogonal closed subspaces H 1, H such that H = H 1 H, and for every x = x 1 + x, x i H i, Ax = x 1 x, i.e., A is a reflection. (ii)= (i): Let A = P I. Then A is obviously self-adjoint, and A A = A = (P I)(P I) = 4P P P + I = I, and thus A is a unitary. (i)= (ii): Let A be a self-adjoint unitary, and define P := 1 (A + I). Then P is obvously self-adjoint, and P = 1 4 ( A + A + I ) = 1 4 (I + A + I) = 1 (A + I) = P, where we used that A = A A = I due to the self-adjointness and unitarity of A. Thus, P is an orthogonal projection. (ii)= (iii): Let H 1 := ran P, H := ker P. Then H 1 H = H, so that every x H can be uniquely decomposed as x = x 1 + x, x i H i. Moreover, we get Ax = (P I)(x 1 + x ) = P (x 1 + x ) (x 1 + x ) = x 1 (x 1 + x ) = x 1 x. (iii)= (ii): Let P be the projection onto H 1. Let x = x 1 + x, x i H i. Then (P I)x = P (x 1 + x ) (x 1 + x ) = x (x 1 + x ) = x 1 x = Ax, and hence A = P I. Scores: Total: 8 points. Each implication above is worth points. (3) a) Show that every partial isometry V B(H) on a finite-dimensional Hilbert space H, can be extended to a unitary on H. b) Show that every bounded operator on a finite-dimensional Hilbert space can be written as the linear combination of at most two unitaries. (Hint: Use the polar decomposition.) 5
6 a) Let V be a partial isometry, i.e., V is an isometry on (ker V ). This implies that r := dim ran V = dim(ker V ). Let e 1,..., e r be an orthonormal basis in (ker V ), and e r+1,..., e d be an orthonormal basis in ker V ; then e 1,..., e r, e r+1,..., e d is an orthonormal basis in the whole Hilbert space H. Similarly, let f 1,..., f r be an orthonormal basis in ran V, f r+1,..., f d be an orthonormal basis in (ran V ) ; then f 1,..., f r, f r+1,..., f d is an orthonormal basis in H. Define { 0, 1 i r, V e i := f i, r + 1 i d. Then (V + V )e i = f i, i = 1,..., r, and hence V + V is an isometric extension of V. b) By the polar decomposition theorem, every bounded operator A can be written as A = V A, where V is a partial isometry. By the solution of Exercise (), A can be written as A = A (U 1 + U ), where U 1, U are unitaries. By the previous part, if the Hilbert space is finite-dimensional then V can be extended to a unitary U, and hence A = A (UU 1 + UU ). By Exercise (1), UU 1 and UU are unitaries, and hence the assertion follows. 6
LECTURE 7. k=1 (, v k)u k. Moreover r
LECTURE 7 Finite rank operators Definition. T is said to be of rank r (r < ) if dim T(H) = r. The class of operators of rank r is denoted by K r and K := r K r. Theorem 1. T K r iff T K r. Proof. Let T
More informationFunctional Analysis Exercise Class
Functional Analysis Exercise Class Week: December 4 8 Deadline to hand in the homework: your exercise class on week January 5. Exercises with solutions ) Let H, K be Hilbert spaces, and A : H K be a linear
More informationLINEAR ALGEBRA BOOT CAMP WEEK 4: THE SPECTRAL THEOREM
LINEAR ALGEBRA BOOT CAMP WEEK 4: THE SPECTRAL THEOREM Unless otherwise stated, all vector spaces in this worksheet are finite dimensional and the scalar field F is R or C. Definition 1. A linear operator
More informationCHAPTER VIII HILBERT SPACES
CHAPTER VIII HILBERT SPACES DEFINITION Let X and Y be two complex vector spaces. A map T : X Y is called a conjugate-linear transformation if it is a reallinear transformation from X into Y, and if T (λx)
More informationMath Linear Algebra II. 1. Inner Products and Norms
Math 342 - Linear Algebra II Notes 1. Inner Products and Norms One knows from a basic introduction to vectors in R n Math 254 at OSU) that the length of a vector x = x 1 x 2... x n ) T R n, denoted x,
More informationI teach myself... Hilbert spaces
I teach myself... Hilbert spaces by F.J.Sayas, for MATH 806 November 4, 2015 This document will be growing with the semester. Every in red is for you to justify. Even if we start with the basic definition
More informationFunctional Analysis Exercise Class
Functional Analysis Exercise Class Week 9 November 13 November Deadline to hand in the homeworks: your exercise class on week 16 November 20 November Exercises (1) Show that if T B(X, Y ) and S B(Y, Z)
More informationHilbert space methods for quantum mechanics. S. Richard
Hilbert space methods for quantum mechanics S. Richard Spring Semester 2016 2 Contents 1 Hilbert space and bounded linear operators 5 1.1 Hilbert space................................ 5 1.2 Vector-valued
More information1 Functional Analysis
1 Functional Analysis 1 1.1 Banach spaces Remark 1.1. In classical mechanics, the state of some physical system is characterized as a point x in phase space (generalized position and momentum coordinates).
More informationSpectral theorems for bounded self-adjoint operators on a Hilbert space
Chapter 10 Spectral theorems for bounded self-adjoint operators on a Hilbert space Let H be a Hilbert space. For a bounded operator A : H H its Hilbert space adjoint is an operator A : H H such that Ax,
More informationADJOINTS, ABSOLUTE VALUES AND POLAR DECOMPOSITIONS
J. OPERATOR THEORY 44(2000), 243 254 c Copyright by Theta, 2000 ADJOINTS, ABSOLUTE VALUES AND POLAR DECOMPOSITIONS DOUGLAS BRIDGES, FRED RICHMAN and PETER SCHUSTER Communicated by William B. Arveson Abstract.
More informationOPERATOR THEORY ON HILBERT SPACE. Class notes. John Petrovic
OPERATOR THEORY ON HILBERT SPACE Class notes John Petrovic Contents Chapter 1. Hilbert space 1 1.1. Definition and Properties 1 1.2. Orthogonality 3 1.3. Subspaces 7 1.4. Weak topology 9 Chapter 2. Operators
More informationAnalysis Preliminary Exam Workshop: Hilbert Spaces
Analysis Preliminary Exam Workshop: Hilbert Spaces 1. Hilbert spaces A Hilbert space H is a complete real or complex inner product space. Consider complex Hilbert spaces for definiteness. If (, ) : H H
More informationNumerical Linear Algebra
University of Alabama at Birmingham Department of Mathematics Numerical Linear Algebra Lecture Notes for MA 660 (1997 2014) Dr Nikolai Chernov April 2014 Chapter 0 Review of Linear Algebra 0.1 Matrices
More informationFunctional Analysis Exercise Class
Functional Analysis Exercise Class Week 2 November 6 November Deadline to hand in the homeworks: your exercise class on week 9 November 13 November Exercises (1) Let X be the following space of piecewise
More informationThe following definition is fundamental.
1. Some Basics from Linear Algebra With these notes, I will try and clarify certain topics that I only quickly mention in class. First and foremost, I will assume that you are familiar with many basic
More informationMath Linear Algebra
Math 220 - Linear Algebra (Summer 208) Solutions to Homework #7 Exercise 6..20 (a) TRUE. u v v u = 0 is equivalent to u v = v u. The latter identity is true due to the commutative property of the inner
More informationThroughout these notes we assume V, W are finite dimensional inner product spaces over C.
Math 342 - Linear Algebra II Notes Throughout these notes we assume V, W are finite dimensional inner product spaces over C 1 Upper Triangular Representation Proposition: Let T L(V ) There exists an orthonormal
More informationTOEPLITZ OPERATORS. Toeplitz studied infinite matrices with NW-SE diagonals constant. f e C :
TOEPLITZ OPERATORS EFTON PARK 1. Introduction to Toeplitz Operators Otto Toeplitz lived from 1881-1940 in Goettingen, and it was pretty rough there, so he eventually went to Palestine and eventually contracted
More informationMath 123 Homework Assignment #2 Due Monday, April 21, 2008
Math 123 Homework Assignment #2 Due Monday, April 21, 2008 Part I: 1. Suppose that A is a C -algebra. (a) Suppose that e A satisfies xe = x for all x A. Show that e = e and that e = 1. Conclude that e
More information2.2. OPERATOR ALGEBRA 19. If S is a subset of E, then the set
2.2. OPERATOR ALGEBRA 19 2.2 Operator Algebra 2.2.1 Algebra of Operators on a Vector Space A linear operator on a vector space E is a mapping L : E E satisfying the condition u, v E, a R, L(u + v) = L(u)
More informationFunctional Analysis II held by Prof. Dr. Moritz Weber in summer 18
Functional Analysis II held by Prof. Dr. Moritz Weber in summer 18 General information on organisation Tutorials and admission for the final exam To take part in the final exam of this course, 50 % of
More information1 Inner Product Space
Ch - Hilbert Space 1 4 Hilbert Space 1 Inner Product Space Let E be a complex vector space, a mapping (, ) : E E C is called an inner product on E if i) (x, x) 0 x E and (x, x) = 0 if and only if x = 0;
More information4.6. Linear Operators on Hilbert Spaces
4.6. Linear Operators on Hilbert Spaces 1 4.6. Linear Operators on Hilbert Spaces Note. This section explores a number of different kinds of bounded linear transformations (or, equivalently, operators
More information08a. Operators on Hilbert spaces. 1. Boundedness, continuity, operator norms
(February 24, 2017) 08a. Operators on Hilbert spaces Paul Garrett garrett@math.umn.edu http://www.math.umn.edu/ garrett/ [This document is http://www.math.umn.edu/ garrett/m/real/notes 2016-17/08a-ops
More informationOn Unitary Relations between Kre n Spaces
RUDI WIETSMA On Unitary Relations between Kre n Spaces PROCEEDINGS OF THE UNIVERSITY OF VAASA WORKING PAPERS 2 MATHEMATICS 1 VAASA 2011 III Publisher Date of publication Vaasan yliopisto August 2011 Author(s)
More informationYour first day at work MATH 806 (Fall 2015)
Your first day at work MATH 806 (Fall 2015) 1. Let X be a set (with no particular algebraic structure). A function d : X X R is called a metric on X (and then X is called a metric space) when d satisfies
More informationUNBOUNDED OPERATORS ON HILBERT SPACES. Let X and Y be normed linear spaces, and suppose A : X Y is a linear map.
UNBOUNDED OPERATORS ON HILBERT SPACES EFTON PARK Let X and Y be normed linear spaces, and suppose A : X Y is a linear map. Define { } Ax A op = sup x : x 0 = { Ax : x 1} = { Ax : x = 1} If A
More informationFall TMA4145 Linear Methods. Solutions to exercise set 9. 1 Let X be a Hilbert space and T a bounded linear operator on X.
TMA445 Linear Methods Fall 26 Norwegian University of Science and Technology Department of Mathematical Sciences Solutions to exercise set 9 Let X be a Hilbert space and T a bounded linear operator on
More informationErrata Applied Analysis
Errata Applied Analysis p. 9: line 2 from the bottom: 2 instead of 2. p. 10: Last sentence should read: The lim sup of a sequence whose terms are bounded from above is finite or, and the lim inf of a sequence
More informationMATH 8253 ALGEBRAIC GEOMETRY WEEK 12
MATH 8253 ALGEBRAIC GEOMETRY WEEK 2 CİHAN BAHRAN 3.2.. Let Y be a Noetherian scheme. Show that any Y -scheme X of finite type is Noetherian. Moreover, if Y is of finite dimension, then so is X. Write f
More informationSPECTRAL THEOREM FOR COMPACT SELF-ADJOINT OPERATORS
SPECTRAL THEOREM FOR COMPACT SELF-ADJOINT OPERATORS G. RAMESH Contents Introduction 1 1. Bounded Operators 1 1.3. Examples 3 2. Compact Operators 5 2.1. Properties 6 3. The Spectral Theorem 9 3.3. Self-adjoint
More information1 Math 241A-B Homework Problem List for F2015 and W2016
1 Math 241A-B Homework Problem List for F2015 W2016 1.1 Homework 1. Due Wednesday, October 7, 2015 Notation 1.1 Let U be any set, g be a positive function on U, Y be a normed space. For any f : U Y let
More information5 Compact linear operators
5 Compact linear operators One of the most important results of Linear Algebra is that for every selfadjoint linear map A on a finite-dimensional space, there exists a basis consisting of eigenvectors.
More informationOctober 25, 2013 INNER PRODUCT SPACES
October 25, 2013 INNER PRODUCT SPACES RODICA D. COSTIN Contents 1. Inner product 2 1.1. Inner product 2 1.2. Inner product spaces 4 2. Orthogonal bases 5 2.1. Existence of an orthogonal basis 7 2.2. Orthogonal
More informationSingular Value Decomposition (SVD) and Polar Form
Chapter 2 Singular Value Decomposition (SVD) and Polar Form 2.1 Polar Form In this chapter, we assume that we are dealing with a real Euclidean space E. Let f: E E be any linear map. In general, it may
More informationSpectral theory for compact operators on Banach spaces
68 Chapter 9 Spectral theory for compact operators on Banach spaces Recall that a subset S of a metric space X is precompact if its closure is compact, or equivalently every sequence contains a Cauchy
More informationSymmetric and self-adjoint matrices
Symmetric and self-adjoint matrices A matrix A in M n (F) is called symmetric if A T = A, ie A ij = A ji for each i, j; and self-adjoint if A = A, ie A ij = A ji or each i, j Note for A in M n (R) that
More informationMeans of unitaries, conjugations, and the Friedrichs operator
J. Math. Anal. Appl. 335 (2007) 941 947 www.elsevier.com/locate/jmaa Means of unitaries, conjugations, and the Friedrichs operator Stephan Ramon Garcia Department of Mathematics, Pomona College, Claremont,
More informationTHE PROBLEMS FOR THE SECOND TEST FOR BRIEF SOLUTIONS
THE PROBLEMS FOR THE SECOND TEST FOR 18.102 BRIEF SOLUTIONS RICHARD MELROSE Question.1 Show that a subset of a separable Hilbert space is compact if and only if it is closed and bounded and has the property
More informationFunctional Analysis I
Functional Analysis I Course Notes by Stefan Richter Transcribed and Annotated by Gregory Zitelli Polar Decomposition Definition. An operator W B(H) is called a partial isometry if W x = X for all x (ker
More informationElementary linear algebra
Chapter 1 Elementary linear algebra 1.1 Vector spaces Vector spaces owe their importance to the fact that so many models arising in the solutions of specific problems turn out to be vector spaces. The
More informationMAT 445/ INTRODUCTION TO REPRESENTATION THEORY
MAT 445/1196 - INTRODUCTION TO REPRESENTATION THEORY CHAPTER 1 Representation Theory of Groups - Algebraic Foundations 1.1 Basic definitions, Schur s Lemma 1.2 Tensor products 1.3 Unitary representations
More informationCompact operators on Banach spaces
Compact operators on Banach spaces Jordan Bell jordan.bell@gmail.com Department of Mathematics, University of Toronto November 12, 2017 1 Introduction In this note I prove several things about compact
More informationLINEAR ALGEBRA QUESTION BANK
LINEAR ALGEBRA QUESTION BANK () ( points total) Circle True or False: TRUE / FALSE: If A is any n n matrix, and I n is the n n identity matrix, then I n A = AI n = A. TRUE / FALSE: If A, B are n n matrices,
More informationChapter 8 Integral Operators
Chapter 8 Integral Operators In our development of metrics, norms, inner products, and operator theory in Chapters 1 7 we only tangentially considered topics that involved the use of Lebesgue measure,
More informationMATH 113 SPRING 2015
MATH 113 SPRING 2015 DIARY Effective syllabus I. Metric spaces - 6 Lectures and 2 problem sessions I.1. Definitions and examples I.2. Metric topology I.3. Complete spaces I.4. The Ascoli-Arzelà Theorem
More informationMath 113 Final Exam: Solutions
Math 113 Final Exam: Solutions Thursday, June 11, 2013, 3.30-6.30pm. 1. (25 points total) Let P 2 (R) denote the real vector space of polynomials of degree 2. Consider the following inner product on P
More informationSpectral Theorem for Self-adjoint Linear Operators
Notes for the undergraduate lecture by David Adams. (These are the notes I would write if I was teaching a course on this topic. I have included more material than I will cover in the 45 minute lecture;
More informationReal Analysis, 2nd Edition, G.B.Folland Elements of Functional Analysis
Real Analysis, 2nd Edition, G.B.Folland Chapter 5 Elements of Functional Analysis Yung-Hsiang Huang 5.1 Normed Vector Spaces 1. Note for any x, y X and a, b K, x+y x + y and by ax b y x + b a x. 2. It
More informationLinear Algebra 2 Spectral Notes
Linear Algebra 2 Spectral Notes In what follows, V is an inner product vector space over F, where F = R or C. We will use results seen so far; in particular that every linear operator T L(V ) has a complex
More informationMATH 423 Linear Algebra II Lecture 33: Diagonalization of normal operators.
MATH 423 Linear Algebra II Lecture 33: Diagonalization of normal operators. Adjoint operator and adjoint matrix Given a linear operator L on an inner product space V, the adjoint of L is a transformation
More informationMA5206 Homework 4. Group 4. April 26, ϕ 1 = 1, ϕ n (x) = 1 n 2 ϕ 1(n 2 x). = 1 and h n C 0. For any ξ ( 1 n, 2 n 2 ), n 3, h n (t) ξ t dt
MA526 Homework 4 Group 4 April 26, 26 Qn 6.2 Show that H is not bounded as a map: L L. Deduce from this that H is not bounded as a map L L. Let {ϕ n } be an approximation of the identity s.t. ϕ C, sptϕ
More informationLecture notes: Applied linear algebra Part 1. Version 2
Lecture notes: Applied linear algebra Part 1. Version 2 Michael Karow Berlin University of Technology karow@math.tu-berlin.de October 2, 2008 1 Notation, basic notions and facts 1.1 Subspaces, range and
More informationj=1 x j p, if 1 p <, x i ξ : x i < ξ} 0 as p.
LINEAR ALGEBRA Fall 203 The final exam Almost all of the problems solved Exercise Let (V, ) be a normed vector space. Prove x y x y for all x, y V. Everybody knows how to do this! Exercise 2 If V is a
More informationLinear Algebra Lecture Notes-II
Linear Algebra Lecture Notes-II Vikas Bist Department of Mathematics Panjab University, Chandigarh-64 email: bistvikas@gmail.com Last revised on March 5, 8 This text is based on the lectures delivered
More informationCOMMON COMPLEMENTS OF TWO SUBSPACES OF A HILBERT SPACE
COMMON COMPLEMENTS OF TWO SUBSPACES OF A HILBERT SPACE MICHAEL LAUZON AND SERGEI TREIL Abstract. In this paper we find a necessary and sufficient condition for two closed subspaces, X and Y, of a Hilbert
More informationThe spectrum of a self-adjoint operator is a compact subset of R
The spectrum of a self-adjoint operator is a compact subset of R Jordan Bell jordan.bell@gmail.com Department of Mathematics, University of Toronto April 3, 2014 Abstract In these notes I prove that the
More informationMath 115A: Homework 5
Math 115A: Homework 5 1 Suppose U, V, and W are finite-dimensional vector spaces over a field F, and that are linear a) Prove ker ST ) ker T ) b) Prove nullst ) nullt ) c) Prove imst ) im S T : U V, S
More informationQuantum Information & Quantum Computing
Math 478, Phys 478, CS4803, February 9, 006 1 Georgia Tech Math, Physics & Computing Math 478, Phys 478, CS4803 Quantum Information & Quantum Computing Problems Set 1 Due February 9, 006 Part I : 1. Read
More informationLecture Notes on Operator Algebras. John M. Erdman Portland State University. Version March 12, 2011
Lecture Notes on Operator Algebras John M. Erdman Portland State University Version March 12, 2011 c 2010 John M. Erdman E-mail address: erdman@pdx.edu Contents Chapter 1. LINEAR ALGEBRA AND THE SPECTRAL
More informationSPECTRAL THEORY EVAN JENKINS
SPECTRAL THEORY EVAN JENKINS Abstract. These are notes from two lectures given in MATH 27200, Basic Functional Analysis, at the University of Chicago in March 2010. The proof of the spectral theorem for
More informationCHAPTER X THE SPECTRAL THEOREM OF GELFAND
CHAPTER X THE SPECTRAL THEOREM OF GELFAND DEFINITION A Banach algebra is a complex Banach space A on which there is defined an associative multiplication for which: (1) x (y + z) = x y + x z and (y + z)
More informationA Brief Introduction to Functional Analysis
A Brief Introduction to Functional Analysis Sungwook Lee Department of Mathematics University of Southern Mississippi sunglee@usm.edu July 5, 2007 Definition 1. An algebra A is a vector space over C with
More informationHomework If the inverse T 1 of a closed linear operator exists, show that T 1 is a closed linear operator.
Homework 3 1 If the inverse T 1 of a closed linear operator exists, show that T 1 is a closed linear operator Solution: Assuming that the inverse of T were defined, then we will have to have that D(T 1
More informationMATH 583A REVIEW SESSION #1
MATH 583A REVIEW SESSION #1 BOJAN DURICKOVIC 1. Vector Spaces Very quick review of the basic linear algebra concepts (see any linear algebra textbook): (finite dimensional) vector space (or linear space),
More informationSelf-adjoint extensions of symmetric operators
Self-adjoint extensions of symmetric operators Simon Wozny Proseminar on Linear Algebra WS216/217 Universität Konstanz Abstract In this handout we will first look at some basics about unbounded operators.
More information10.1. The spectrum of an operator. Lemma If A < 1 then I A is invertible with bounded inverse
10. Spectral theory For operators on finite dimensional vectors spaces, we can often find a basis of eigenvectors (which we use to diagonalize the matrix). If the operator is symmetric, this is always
More informationDavid Hilbert was old and partly deaf in the nineteen thirties. Yet being a diligent
Chapter 5 ddddd dddddd dddddddd ddddddd dddddddd ddddddd Hilbert Space The Euclidean norm is special among all norms defined in R n for being induced by the Euclidean inner product (the dot product). A
More informationThe Dirichlet-to-Neumann operator
Lecture 8 The Dirichlet-to-Neumann operator The Dirichlet-to-Neumann operator plays an important role in the theory of inverse problems. In fact, from measurements of electrical currents at the surface
More informationOPERATOR THEORY - PART 3/3. Contents
OPERATOR THEORY - PART 3/3 T.K.SUBRAHMONIAN MOOTHATHU Contents 1. Approximate eigenvalues, and another polar decomposition 1 2. The spectrum of a compact operator 5 3. Index theory and compact operators
More informationFunctional Analysis. Franck Sueur Metric spaces Definitions Completeness Compactness Separability...
Functional Analysis Franck Sueur 2018-2019 Contents 1 Metric spaces 1 1.1 Definitions........................................ 1 1.2 Completeness...................................... 3 1.3 Compactness......................................
More informationCHAPTER V DUAL SPACES
CHAPTER V DUAL SPACES DEFINITION Let (X, T ) be a (real) locally convex topological vector space. By the dual space X, or (X, T ), of X we mean the set of all continuous linear functionals on X. By the
More informationClassical stuff - title to be changed later
CHAPTER 1 Classical stuff - title to be changed later 1. Positive Definite Kernels To start with something simple and elegant, we choose positive definite kernels which appear at every corner in functional
More informationMathematics Department Stanford University Math 61CM/DM Inner products
Mathematics Department Stanford University Math 61CM/DM Inner products Recall the definition of an inner product space; see Appendix A.8 of the textbook. Definition 1 An inner product space V is a vector
More informationREAL ANALYSIS II HOMEWORK 3. Conway, Page 49
REAL ANALYSIS II HOMEWORK 3 CİHAN BAHRAN Conway, Page 49 3. Let K and k be as in Proposition 4.7 and suppose that k(x, y) k(y, x). Show that K is self-adjoint and if {µ n } are the eigenvalues of K, each
More informationReview problems for MA 54, Fall 2004.
Review problems for MA 54, Fall 2004. Below are the review problems for the final. They are mostly homework problems, or very similar. If you are comfortable doing these problems, you should be fine on
More informationNotes on Banach Algebras and Functional Calculus
Notes on Banach Algebras and Functional Calculus April 23, 2014 1 The Gelfand-Naimark theorem (proved on Feb 7) Theorem 1. If A is a commutative C -algebra and M is the maximal ideal space, of A then the
More informationReal Variables # 10 : Hilbert Spaces II
randon ehring Real Variables # 0 : Hilbert Spaces II Exercise 20 For any sequence {f n } in H with f n = for all n, there exists f H and a subsequence {f nk } such that for all g H, one has lim (f n k,
More informationMAT 578 FUNCTIONAL ANALYSIS EXERCISES
MAT 578 FUNCTIONAL ANALYSIS EXERCISES JOHN QUIGG Exercise 1. Prove that if A is bounded in a topological vector space, then for every neighborhood V of 0 there exists c > 0 such that tv A for all t > c.
More informationChapter 4 Euclid Space
Chapter 4 Euclid Space Inner Product Spaces Definition.. Let V be a real vector space over IR. A real inner product on V is a real valued function on V V, denoted by (, ), which satisfies () (x, y) = (y,
More informationON THE SIMILARITY OF CENTERED OPERATORS TO CONTRACTIONS. Srdjan Petrović
ON THE SIMILARITY OF CENTERED OPERATORS TO CONTRACTIONS Srdjan Petrović Abstract. In this paper we show that every power bounded operator weighted shift with commuting normal weights is similar to a contraction.
More informationHilbert Spaces. Hilbert space is a vector space with some extra structure. We start with formal (axiomatic) definition of a vector space.
Hilbert Spaces Hilbert space is a vector space with some extra structure. We start with formal (axiomatic) definition of a vector space. Vector Space. Vector space, ν, over the field of complex numbers,
More informationLinear Algebra and Dirac Notation, Pt. 2
Linear Algebra and Dirac Notation, Pt. 2 PHYS 500 - Southern Illinois University February 1, 2017 PHYS 500 - Southern Illinois University Linear Algebra and Dirac Notation, Pt. 2 February 1, 2017 1 / 14
More informationOrthogonal complement
Orthogonal complement Aim lecture: Inner products give a special way of constructing vector space complements. As usual, in this lecture F = R or C. We also let V be an F-space equipped with an inner product
More informationFunctional Analysis, Math 7321 Lecture Notes from April 04, 2017 taken by Chandi Bhandari
Functional Analysis, Math 7321 Lecture Notes from April 0, 2017 taken by Chandi Bhandari Last time:we have completed direct sum decomposition with generalized eigen space. 2. Theorem. Let X be separable
More informationRIESZ BASES AND UNCONDITIONAL BASES
In this paper we give a brief introduction to adjoint operators on Hilbert spaces and a characterization of the dual space of a Hilbert space. We then introduce the notion of a Riesz basis and give some
More informationSpectral Theory, with an Introduction to Operator Means. William L. Green
Spectral Theory, with an Introduction to Operator Means William L. Green January 30, 2008 Contents Introduction............................... 1 Hilbert Space.............................. 4 Linear Maps
More informationHILBERT SPACES AND THE RADON-NIKODYM THEOREM. where the bar in the first equation denotes complex conjugation. In either case, for any x V define
HILBERT SPACES AND THE RADON-NIKODYM THEOREM STEVEN P. LALLEY 1. DEFINITIONS Definition 1. A real inner product space is a real vector space V together with a symmetric, bilinear, positive-definite mapping,
More information2.2. Show that U 0 is a vector space. For each α 0 in F, show by example that U α does not satisfy closure.
Hints for Exercises 1.3. This diagram says that f α = β g. I will prove f injective g injective. You should show g injective f injective. Assume f is injective. Now suppose g(x) = g(y) for some x, y A.
More informationvon Neumann algebras, II 1 factors, and their subfactors V.S. Sunder (IMSc, Chennai)
von Neumann algebras, II 1 factors, and their subfactors V.S. Sunder (IMSc, Chennai) Lecture 3 at IIT Mumbai, April 24th, 2007 Finite-dimensional C -algebras: Recall: Definition: A linear functional tr
More informationSummary of Week 9 B = then A A =
Summary of Week 9 Finding the square root of a positive operator Last time we saw that positive operators have a unique positive square root We now briefly look at how one would go about calculating the
More informationChapter 1 Foundations of Elliptic Boundary Value Problems 1.1 Euler equations of variational problems
Chapter 1 Foundations of Elliptic Boundary Value Problems 1.1 Euler equations of variational problems Elliptic boundary value problems often occur as the Euler equations of variational problems the latter
More informationGaussian automorphisms whose ergodic self-joinings are Gaussian
F U N D A M E N T A MATHEMATICAE 164 (2000) Gaussian automorphisms whose ergodic self-joinings are Gaussian by M. L e m a ńc z y k (Toruń), F. P a r r e a u (Paris) and J.-P. T h o u v e n o t (Paris)
More informationProblem # Max points possible Actual score Total 120
FINAL EXAMINATION - MATH 2121, FALL 2017. Name: ID#: Email: Lecture & Tutorial: Problem # Max points possible Actual score 1 15 2 15 3 10 4 15 5 15 6 15 7 10 8 10 9 15 Total 120 You have 180 minutes to
More informationFunctional Analysis HW #5
Functional Analysis HW #5 Sangchul Lee October 29, 2015 Contents 1 Solutions........................................ 1 1 Solutions Exercise 3.4. Show that C([0, 1]) is not a Hilbert space, that is, there
More informationMathematics Department Stanford University Math 61CM/DM Vector spaces and linear maps
Mathematics Department Stanford University Math 61CM/DM Vector spaces and linear maps We start with the definition of a vector space; you can find this in Section A.8 of the text (over R, but it works
More information11. Spectral theory For operators on finite dimensional vectors spaces, we can often find a basis of eigenvectors (which we use to diagonalize the
11. Spectral theory For operators on finite dimensional vectors spaces, we can often find a basis of eigenvectors (which we use to diagonalize the matrix). If the operator is symmetric, this is always
More informationKernel Method: Data Analysis with Positive Definite Kernels
Kernel Method: Data Analysis with Positive Definite Kernels 2. Positive Definite Kernel and Reproducing Kernel Hilbert Space Kenji Fukumizu The Institute of Statistical Mathematics. Graduate University
More informationALGEBRAIC GEOMETRY COURSE NOTES, LECTURE 2: HILBERT S NULLSTELLENSATZ.
ALGEBRAIC GEOMETRY COURSE NOTES, LECTURE 2: HILBERT S NULLSTELLENSATZ. ANDREW SALCH 1. Hilbert s Nullstellensatz. The last lecture left off with the claim that, if J k[x 1,..., x n ] is an ideal, then
More information