Content of the lecture
October 17th
Introduction; practical information; requirements for the
grade. Please sign
up for the Exercise class!
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.
October 18th
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)).
Definition of compact operators; discussion and remarks. 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 (see main page at the end).
Algebraic properties of adjoints.
October 24th
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'.
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 for T. Talk about 'pertubation results'. Corollary: The
subset of B(X) of
invertible operators is
an open set.
Discussion of computing functions of an operator: monomials,
polynomials, (complex)
power series.
October 25th
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 of the identity (A = I
- T, T compact) is a Fredholm operator of index 0.
Theorem: Spectral Theorem for compact operators
(Riesz-Schauder). (Proof starting next time, using the theorem above.)
October 31th
Proof of the the first 2 points 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), and start on point 3 (on Riesz decomposition).
November 7th
Conclusion of the proof of the Riesz-Schauder theorem. Riesz
decomposition.
Back to the spectral theory of Fredholm operators of the for 1-T, T
compact.
1-T (T compact) has finite-dim kernel, closed range, if it's injective
it's also invertible, codim(range) = dim(kernel).
Consequence: the Fredholm alternative. Either Tz-ax=y has a unique sol,
or Tx-ax=0 have finitely many non-trivial, linearly indep soll.
Discussion: the flavour how Fredholm applies to PDE theory and to other
problems. (Some are forthcoming homework!)
November 8th
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 (with proof).
Finite-dimensional case: block decomposition. Jordan normal form.
Schauder Theorem: T is compact iff T' is (with proof). Mentioned how the
proof drastically simplifies if the space is Hilbert.
Outlook of what is coming next: for compact+normal ops we will be able
to give a full decomposition of the H-space in eigenspaces.
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.
November 14th
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, equality holds.
November 15th
Definition of positive semi-definite bounded operator.
The spectrum of a bounded selfadjoint 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 operators on a complex
Hilbert space and beginning of the proof (to be finished next week).
November 21st
End of proof of the Spectral Theorem for normal operators on a complex
Hilbert space.
Spectral Theorem for normal 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.
November 22nd
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 selfadjoint operators.
Motivation: Diagonalisation, thought of in different ways (unitary
equivalence to multiplication operator; expressed by weighted sum of
projection operators). Usefull for computing functions of
operators.
Continuous Functional Calculus for bounded selfadjoint operator T: Statement
and discussion. Uniqueness: Follows from Weierstrass' Approximation
Theorem (not proved yet). Existence (next time): Will also follow from this.
November 28th
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 selfadjoint operator T.
Further properties of the Continuous Functional Calculus for bounded
selfadjoint operator T, and proofs (missing: Proof of Spectral Mapping
Theorem - next time).
November 29th
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 normal 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.
Start with preliminaries on Lebesque integration theory (see crash
course in FA1, and literature - ex. Elstrodt 'Mass-und
Integrationstheorie' or Rudin 'Real and Complex Analysis' or appendix in Werner).
December 5th
Continuation of definitions/results/recall on Lebesgue integration
theory: 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' 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 the
integration against a (unique) complex measure. This one-to-one
correspondance is an isometry. (No proofs; see Rudin, Werner, Elstodt).
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 (proof later).
For a selfadjoint operatoron 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 of
uniqueness. Start on proof of existence: Lax-Milgram Theorem (proof in
tutorial).
December 6th
Proof of Measurable Functional Calculus.
Proof of 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).
December 12th
End of proof of lemma (see Dec 6th).
Lemma: Using the measurable functional calculus for characteristic
functions (of Borel sets) gives orthogonal projections.
Properties.
This gives a special case of a spectral measure; definition of
spectral measure (also called projection valued measure). Compact
support of such measures. Integration with respect to such measures:
For characteristic functions, then step functions, then general
bounded, measurable functions.
December 13th
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).
December 19th
End of proof (E_{sp(T)} = Id; 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.
December 20th
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: after Christmas).
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: after Christmas).
January 9th
Proof of Spectral Theorem, multiplication operator - version, and discussions.
(Very) Brief discussion of Spectral Theorem for normal bounded operators.
January 10th
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).
January 16th
Definition of closed operator and connection to continuity. Closed Graph Thm (reminder).
Thm: 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.
Cor: T is sym iff T^* is an ext of T. In this case, T subset T^**
subset T^* = T^***. T is closed and symmetric iff T=T^** subset T^*. T
= T^* iff T = T^** = T^*.
Def. of essentially selfdjoint. Discussion of selfadjoint
extensions.
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. (Rest of proof next time).
January 17th
Finish proof of Lemma last time.
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.
(Proof of "if" next time).
Definition of resolvent set, resolvent map, spectrum, eigenvalue,
eigenvector. Discussions (!). In particular, spectrum is all of C if T NOT closed operator.
Prop: 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.
January 23th
Other half of missing proof from last time (see above).
Chapter 4: Spectral theory for unbounded self adjoint operators.
Prop: The spectrum of a self adjoint operator is real.
Spectral thm for for unbounded self adjoint operators, multiplication operator version (end of proof next time).
January 24th
Spectral thm for for unbounded self adjoint operators, multiplication operator version (end of proof).
Spectral decompostion for unbounded s.a. operators (end of proof next time).
January 30th
End of proof of Spectral Theorem (in form of Spectral decompostion for unbounded s.a. operators).
Corollary: Measurable Functional calculus (for measurable _bounded_
functions) for selfadjoint unbounded operator. (Proof contained in
proof of Spectral Thm).
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). (Proof: Next time).
(Questionnaires/Umfrage about feedback on the Lecture were handed
out. PLEASE fill them in, and hand them back in the next (!)
Lecture, in the next (!) Uebung, or where you hand in your homework).
January 31th
Proof of corollary mentioned above. Discussion.
Discussion of alternativ proof-strategy of the spectral thm (for
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.
February 6th
Further outlooks: Banach algebra setup: Definition and examples of
Banach algebras. Definition of invertability, spectrum, spectrum
radius. Some results (without proofs).
February 7th
Continuation of outlooks on Banach algebras: Riesz/Dunford-Schwarz
(holomorphic) functional calculus.
Discussion (_very_ brief) of nuclear operators.
Last update: February 7th., 2012 by Thomas Østergaard Sørensen.