### Vorlesung: Funktionalanalysis II (FA2) (WiSe 2015/16)

Content of the lecture (Kurzübersicht der Vorlesung):

13 October:

Introduction, practical information (see main page).

Chapter 0: Motivation and repetition.

Banach space, Hilbert space, examples, bounded linear operators, bounded linear functionals, dual spaces. The dual of a Hilbert space H is "equal" to H via anti-linear isometry Phi. Bi-dual and canonical embedding. Reflexive spaces. Hilbert spaces are all reflexive; other examples.

Definition of resolvent set, spectrum, point spectrum, continuous spectrum, rest/residual spectrum. Resolvent, and resolvent map. Eigenvalues (point spectrum), eigenvectors, and eigenspaces. Invariant subspaces.

The resolvent set is an open subset of C, and the resolvent map is a complex analytic map from the resolvent set to B(X) (bounded operators). Discussion on the aim of the course: The study of spectra, and generalizations of the spectral theorem for symmetric/self-adjoint matrices, to various classes of operators (compact, self-adjoint (bounded/un-bounded)).

14 October:

Definition of compact operators.

Discussion and remarks on compact operators. Integral operators are typical compact operators.

Definition of the (Banach) adjoint of a bounded operator. Definition of Hilbert space adjoint of bounded operator, and of self-adjoint-ness. Properties of adjoints. Programme for the course.

For T in B(X,Y) (X,Y Banach): T is invertible iff T' is invertible (and then (T^{-1})'=(T')^{-1}). So 0 is in the spectrum of T iff it is in the spectrum of T'. End of proof next time.

20 October:

End of proof from last time (see above).

Lemma needed in proof: N(T') = Annil(R(T)) (set of bounded linear functionals on Y vanishing identically on R(T)). Proposition needed in proof (consequence of Hahn-Banach): Y closed linear subspace of normed space X, and x in X but not in Y. Then there exists x' in X' such that (s.t.) x' = 0 on Y, ||x'||=1, and x'(x)=dist(x,Y).

If T in B(X) with ||T|| < 1 then I - T is invertible and its inverse is given by the Neumann series for T. So ||T|| < 1 implies 1 is in the resolvent set for T. Discussion of 'pertubation results'.
Corollary: The subset of B(X) of invertible operators is an open set.

Discussion of computing functions of a bounded operator: monomials, polynomials, (complex) power series.

Examples of functions (given by power series) defined on T in B(X) (exp(T), ...)

21 October:

Continued: Examples of functions (given by power series) defined on T in B(X) (exp(T), log(T), ...)

Chapter 1: Spectral theory for compact operators.

Discussion of solving the equation Tx - lambda x = y, and of 'degrees of freedom' (number of solutions) and 'constraints' (related to existence of solutions).

Definition of Fredholm operator and its index. Discussions.

Theorem (proof later): Any compact pertubation A of the identity I (A = I - T, T compact) is a Fredholm operator of index 0.

Theorem: Spectral Theorem for compact operators (Riesz-Schauder). (Proof using the theorem above.)

Proof of the first point, and some of the proof of the second point (to be finished next time), in the Spectral Theorem for compact operators (spectrum consists of eigenvalues, and possibly 0, is countable, and has zero as only possible accumulation point, and that the order of any eigenvalue is finite).

27 October:

End of proof of the Spectral Theorem for compact operators.

28 October:

Back to the spectral theory of Fredholm operators of the form 1 - T, T compact:
Theorem: If A = 1 - T, T compact, then A is Fredholm, with index 0. A has finite-dim kernel, closed range, if A is injective it is also invertible, codim(range) = dim(kernel). (End of proof next time.)

03 November:

End of proof from last time.

Consequence: The Fredholm alternative: For T compact, a neq 0: Either Tx-ax=y has a unique sol, or Tx-ax=0 have finitely many non-trivial, linearly indep sol's.

Consequence of the Spectral Theorem for compact operators. Non-zero eigenvalues of a compact T are isolated poles of the resolvent of T with order dictated by Riesz-Schauder (without proof).

Finite-dimensional case: block decomposition.

Jordan normal form.

Schauder Theorem: T is compact iff T' is (without proof - see FA1).

Outlook of what is coming next: for compact+normal operators we will be able to give a full decomposition of the H-space in eigenspaces.

04 November:

Def of normal operators on a H-space: T is normal iff T and T* commute. Equivalently, iff Tx and T*x have the same norm for all x.

For a bounded operator on a Banach space, the spectrum is compact, and the spectral radius equals the limit of ||T^m||^{1/m}, and is smaller equal norm of T.

Proof needs Cauchy's Integral Formula for analytic maps with values in a Banach space (and hence the Riemann integral for maps of a real variable with values in a Banach space).

For a _normal_ operator on a complex Hilbert space, the spectral radius (equals the limit of ||T^m||^{1/m}, and) is _equal_ norm of T. (Proof next time.)

10 November:

Proof from last time (see above).

Definition of positive semi-definite bounded operator.

The spectrum of a bounded self-adjoint operator is real and contained in the interval from -||T|| to ||T||. If T is also compact, then either -||T|| or ||T|| (or both) is an eigenvalue. The spectrum of a positive semi-definite bounded operator is contained in the interval from 0 to ||T||. If T is also compact, then ||T|| is an eigenvalue.

Discussion of computing the largest eigenvalue (and hence, consecutive eigenvalues) of a positive semi-definite, compact operator by the optimisation problem: Maximize under the constraint ||x|| = 1. Discussion of the finite dimensional case and alternative to computing zeros of determinants.

Formulation of the Spectral Theorem for normal compact operators on a complex Hilbert space and beginning of the proof (to be finished next time).

11 November:

End of proof of the Spectral Theorem for normal compact operators on a complex Hilbert space.

Spectral Theorem for normal compact operators on a complex Hilbert space, Projection Version.

Definition of (unique) positive, compact square root of a positive compact operator.

Definition of |T| for T a compact operator between two (complex) Hilbert spaces.

Polar decomposition T=U|T| of compact operator T between two (complex) Hilbert spaces.

17 November:

Singular Value Decomposition of compact operator between two (complex) Hilbert spaces.

Outlook: Where do compact operators occur (ex. integral operators and embeddings); brief (!) discussion of boundary value problems (Dirichlet problem); Schatten classes; nuclear operators between Banach spaces.

Chapter 2: Spectral theory for bounded self-adjoint operators.

Motivation: Diagonalisation, thought of in different ways: unitary equivalence to multiplication operator; to be continued next time.

18 November:

Motivation (continued last time): Diagonalisation, thought of in different ways: expressed by weighted sum of projection operators. Useful for computing functions of operators.

Continuous Functional Calculus for bounded self-adjoint operator T: Statement and discussion. Uniqueness: Follows from Weierstrass' Approximation Theorem (no proof here). Existence: Will also follow from this.

Discussion of Weierstrass' Approximation Theorem.

Tietze-Urysohn's Extension Theorem (for continuous functions on closed subsets of metric spaces).

Proof of Continuous Functional Calculus for bounded self-adjoint operator T.

24 November:

Further properties of the Continuous Functional Calculus for bounded self-adjoint operator T, and proofs, including:

Proof of Spectral Mapping Theorem (for continuous functions).

Discussion (sketchy/imprecise and no proofs) of Holomophic Functional Calculus (a la Dunford-Schwartz 'Linear Operators Part I' pp. 568) for general bounded operators on a complex Banach (!) space, and holomorphic functions on (!) its spectrum.

Discussion of relationship between Spectral Theorem/Spectral Decomposition for s.a. compact operators, and the Continuous Functional Calculus in this case: f(T) can be calculated via the spectral decomposition - and the spectral projectors can be found via the calculus. Discussion of problem at zero, and, by analogy: The problem of constructing projections (for instance eigenprojections) for general bounded (s.a.) operators via just _continuous_ functions. Solution: Extend calculus to measurable functions.

25 November:

Start with preliminaries (definitions/results/recall) on Lebesque integration theory (see literature - ex. Elstrodt 'Mass-und Integrationstheorie' or Rudin 'Real and Complex Analysis' or appendix in Werner):
Dynkin systems, measurable functions, step functions, properties.
Signed/complex measures on a set T, the set M(T,Sigma) of all complex measures, the total variation ||mu|| of signed measure mu. (M(T), || ||) is a Banach space.
Riesz'/Riesz-Markov-Kakutani Representation Theorem: Any bounded linear functional on the space of (complex valued) continuous functions on a compact topological space (with the sup-norm) is given by integration against a (unique) complex measure. This one-to-one correspondance is an isometry. (No proofs; see Rudin, Werner, Elstrodt).

Lemma on the space of measurable functions on a compact subset of C: This is the smallest function space which contains all continuous functions and is stable under pointwise limits of uniformly bounded sequences. (End of proof next time).

01 December:

End of proof of Lemma from last time (see above).

For a self-adjoint operator on Hilbert space H, and two (fixed) x,y in H, the continuous linear functional calculus gives a unique measure mu_{x,y} (by Riesz' Repr Thm above). The map assigning mu_{x,y} to (x,y) is sesquilinear and bounded.

Statement of the Measurable Functional Calculus. Proof next time.

02 December:

Theorem: Measurable Functional Calculus. Proof of uniqueness. Proof of existence (uses: Lax-Milgram Theorem (no proof)).

Lemma: Using the Measurable Functional Calculus for characteristic functions (of Borel sets) gives orthogonal projections. Properties.

Lemma: This gives a special case of a spectral measure; definition of spectral measure (also called projection valued measure). Compact support of such measures.

08 December:

Integration with respect to spectral measures: For characteristic functions, then step functions, then general bounded, measurable functions.

The integration with respect to a spectral measure delivers a bounded linear map from the space of measurable bounded functions to the space of bounded operators. Real functions give self-adjoint operators.

For a spectral measure with compact support, one can integrate the function t |-> t to get a self-adjoint bounded operator T.
Thm: Integration with respect to the spectral measure of T (defined as above) is exactly the measurable functional calculus of T. (End of proof next time).

09 December:

End of proof (E_{sp(T)} = I; discussion of regular Borel measures).

Spectral Theorem for self-adjoint bounded operators: For such an operator T, there is a unique spectral measure such that T is given by the integral of the function t|-> t wrt. to the spectral measure. Integration wrt. this measure gives the functional calculus.

15 December (Lecture 1):

Corollary: A bounded operator S commutes with a self-adjoint bounded operator T iff it commutes with all the spectral projections of T.

Examples of what Spectral Thm concrete gives: Finite dim case, compact self-adjoint operator, multiplication operator t on L^2[0,1].

Thm: Characterization of the (real part of) the resolvent set, and of eigenvalues, of a bounded self-adjoint operator via its spectral projections.

Discussion of Spectral Thm in finite dimensions: T is unitarily equivalent to a multiplication operator.

Thm: Any bounded self-adjoint operator WITH a cyclic vector x_0 is unitarily equivalent to the multiplication operator given by multiplication by the variable t, on the L^2 space for the measure given by the spectral measure, and x_0. (Proof next time)

Spectral Theorem, Multiplication Operator Version: Any bounded self-adjoint operator is unitarily equivalent to a multiplication operator on _some_ L^2-space (if H is separable, of a sigma-finite measure). (Proof next time).

15 December (Lecture 2):

Proof of Spectral Theorem (bounded, s.a. operator), Multiplication Operator Version (see above) and discussions.

(Very) Brief discussion of Spectral Theorem for normal bounded operators.

16 December:

Chapter 3: Unbounded operators (in particular, symmetric operators and quadratic forms).

Motivation. Definition of operator T IN a Hilbert space. Extension; equality of operators; symmetric operator. The Hellinger-Toeplitz Thm and discussion.

Definition of adjoint operator T^* of a densely defined operator T and of self-adjointness (T=T^*). Discussions - in particular: s.a. implies symmetric, but not necessarily the other way around. The adjoint T^* however is always an extension of T if T is symmetric (and densely defined). In particular, T^** is well-defined in this case (if not, D(T^*) might be too small).

Definition of closed operator and connection to continuity. Closed Graph Thm (reminder).

Theorem: For a densely defined operator T: T^* is closed. If T^* is densely defined, then T^** is an extension of T, and it is the smallest closed extension of T. (Proof next time).

Merry Christmas, Happy Hanukkah, & Happy Newyear!

12 January:

Proof of theorem from last time (see above).

Corollary: T is sym iff T^* is an extension of T. In this case, T subset T^** subset T^* = T^***. T is closed and symmetric iff T=T^** subset T^*. T = T^* (T s.a.) iff T = T^** = T^*.

Lemma: T densely defined, then N(T^* +/- i) equals the orthogonal complement of R(T -/+ i). If T is symmetric and closed, then R(T +/- i) is closed in H. (End of proof next time).

13 January:

End of proof from last time (see above).

Thm: T is s.a. iff it is closed and N(T^* +/- i) = {0} iff R(T +/- i) = H.

Cor: T is ess. s.a. iff N(T^* +/- i) = {0} iff R(T +/- i) subset H is dense.

Def. of deficiency indices. Thm: A symmetric and densely defined operator has a s.a. extension iff its two deficiency indices are equal. (End of proof next time).

15 January:

End of proof from last time (see above).

Definition of resolvent set, resolvent map, spectrum, eigenvalue, eigenvector.

Discussions (!). In particular, spectrum is all of C if T NOT closed operator.

Proposition: The resolvent set is an open subset of C, hence the spectrum is a closed subset of C. The resolvent map is analytic, and satisfies the resolvent identity. Discussions.

Chapter 4: Spectral theory for unbounded self-adjoint operators.

Proposition: The spectrum of a self-adjoint operator is real.

Spectral Thm for unbounded self-adjoint operators, Multiplication Operator version. (End of proof next time).

19 January:

End of proof from last time (see above).

Spectral decompostion for unbounded s.a. operators (continuation of proof next time).

20 January:

Continuation of proof of Spectral Theorem (in form of Spectral decompostion for unbounded s.a. operators).

Corollary: Measurable Functional Calculus (for measurable _bounded_ functions) for self-adjoint unbounded operator. (Proof contained in proof of Spectral Thm).

End of proof of Spectral Theorem (in form of Spectral decompostion for unbounded s.a. operators).

Corollary: ||h(T)x||^2 is the integral of |h|^2 wrt. the (positive) measure d < x,E_lambda x>. For a sequence of bounded, measurable functions h_n(t), converging pointwise to the function t, and bounded (in absolute value) by |t| (for all t and n), we have that h_n(T)x converges to Tx for any x in D(T). (End of proof next time).

26 January:

End of proof from last time (see above).

Chapter 5: Outlooks.

Discussion of alternativ proof-strategy of the Spectral Thm (for bounded/unbounded s.a. operators): Constructing and studying the Herglotz/Nevanlinna functions F_x(z) = < x, R_T(z) x >, and use complex analysis (Borel transform, Stieltjes inversion formula) to recover the spectral measure (see book by Teschl).

Brief discussion of 'resolutions of the identity' (Spektralschar) and the Riemann-Stieltjes integral of such resolutions.
(See the book by Weidmann - or the German version, Volume 1 and Volume 2. For more on Riemann-Stieltjes integrals, see the book by Stroock.)

27 January:

Outlook: 'Practical' use of the Spectral Theorem via (integral) representations of functions (of various classes):
Cauchy's integral formula, Laplace transform, Fourier transform, Mellin transform. Discussion of time-dependent Schrödinger equation, and 'heat equation'. Other useful (?!?) integral formulae.

02 February:

Outlook: Typical examples of (unbounded) operators: PDOs.

Boundary Value Problems (BVP) or PDOs in all of space.

Fourier transform: Unitary on L^2 which diagonalises every linear Partial Differential Operator (PDO) with _constant_ coefficients. Gives spectrum of ('free') Laplacian (in R^n). Domain of free Laplacian: Sobolev spaces.

Trailer: Course next semester by Cuenin on 'Pseudodifferential Operators' (ΨDOs): Discussion of 'tempered distributions', Sobolev spaces, ΨDOs and their action on these spaces.

Trailer: Course next semester by Seifert on 'Distribution theory and Sobolev spaces'.

03 February:

(Very!) brief discussion of Schrödinger operators H = - Δ + V(x), spectral questions, and 'perturbation theory': Domain, self-adjointness (operators x and p do not commute, so not simultaneously diagonalise/construct functional calculus; related: Weyl-calculus in Cuenin's course).

Spectral questions: Type of spectrum? (eigenvalues/continuous spectrum). 'Typical' spectra: Harmonic oscillator, Hydrogen, Dirichlet/Neumann Laplacians on bounded domain. Questions from Spectral Geometry (e.g. "Can you hear the shape of a drum?"). Discussion of asymptotic distribution of eigenvalues (Weyl's Law/Weyl-asymptotics, semi-classical analysis, Lieb-Thirring inequalities).

(See also 'Weyl’s Law: Spectral Properties of the Laplacian in Mathematics and Physics', W. Arendt, R. Nittka, W. Peter, F. Steiner.
In 'Mathematical Analysis of Evolution, Information, and Complexity.' Edited by Wolfgang Arendt and Wolfgang P. Schleich Copyright © 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 978-3-527-40830-6)

End of lectures!

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Letzte Änderung: 04 February 2016 (no more updates).

Thomas Østergaard Sørensen

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