Functional Analysis (Operator Theory). Spring 2020
Syllabus
- Lecture 1 (17.01.2020).
- Fredholm operators. The Fredholm index. Examples.
The additivity of the index. Kato's lemma (the image of a Fredholm operator is closed).
Some results from the duality theory for Banach spaces (a survey). The Closed Image Theorem.
A corollary: the Fredholm property and index of the dual operator.
- Lecture 2 (24.01.2020).
- Basic facts on compact operators (a survey).
The Riesz-Schauder theory: Riesz's theorem on operators "1+compact", Fredholm alternative,
abstract Fredholm theorems in Schauder's form, properties of the spectrum of a compact operator.
- Lecture 3 (31.01.2020).
- Topological direct sums and complemented subspaces.
Characterizations of complemented subspaces. Finite-dimensional subspaces and closed subspaces
of finite codimension are complemented. The Nikolski-Atkinson criterion for Fredholm operators.
- Lecture 4 (07.02.2020).
- The Calkin algebra. The essential spectrum of a linear
operator. The set of Fredholm operators is open, and the index is locally constant.
The stability of the index under compact perturbations. Nikolski's characterization of Fredholm operators
of index zero. Toeplitz operators on the Hardy space. The index formula for Toeplitz operators.
- Lecture 5 (14.02.2020).
- Topological vector spaces. The topology generated by a family
of seminorms. Locally convex spaces. Examples of locally convex spaces (spaces of continuous and smooth
functions, the Schwartz space, the strong operator topology, the weak operator topology).
The continuity criterion for a seminorm on a locally convex space. The continuity criterion for
a linear operator between locally convex spaces.
- Lecture 6 (21.02.2020).
- Convex, circled, and absolutely convex sets in vector spaces.
Convex, circled, and absolutely convex hulls. Absorbing sets. The Minkowski functional and its properties.
Locally convex spaces are "polynormable". The domination relation for families of seminorms.
Equivalent families of seminorms. Examples.
- Lecture 7 (28.02.2020).
- Linear functionals on locally convex spaces (extension
from subspaces, separation of points, separation of points and subspaces). Dual pairs of vector spaces.
The weak topology of a dual pair. Special cases and basic properties of the weak topology.
A characterization of linear functionals continuous with respect to the weak topology.
A characterization of reflexive Banach spaces in terms of topologies on the dual.
Duality of operators between dual pairs. A relation to weak continuity.
- Lecture 8 (06.03.2020).
- Annihilators and polars. Basic properties of annihilators
and polars. The bipolar theorem (for dual pairs and for locally convex spaces). Corollaries:
the closure of an absolutely convex set equals the weak closure; the double annihilator theorem;
a density criterion for a vector subspace; relations between kernels and images of a linear operator
and of the dual operator. Goldstine's theorem. Equicontinuous families of linear operators.
The Banach-Alaoglu-Bourbaki theorem. A characterization of reflexive Banach spaces in terms of weak
compactness.
- Lecture 9 (27.03.2020) (video)
- The maximal spectrum
and the character space of a commutative unital algebra.
The closedness of maximal ideals of a commutative unital Banach algebra. The 1-1 correspondence between the character space
and the maximal spectrum of a commutative unital Banach algebra.
The Gelfand topology on the maximal spectrum. The compactness of the maximal spectrum (the unital case).
The Gelfand transform of a commutative unital Banach algebra. The maximal spectrum and the Gelfand transform
for subalgebras of C(X).
- Lecture 10 (05.04.2020) (video)
(lecture notes)
- Banach *-algebras.
C*-algebras. Examples. The spectral radius of a normal element. Corollaries: the uniqueness of the C*-norm on a *-algebra;
the automatic continuity of *-homomorphisms. The spectrum of a selfadjoint element is real. Corollaries:
characters of a C*-algebra are *-characters; the spectral invariance of C*-subalgebras. The 1st (commutative) Gelfand-Naimark theorem.
Functorial properties of the Gelfand transform (the adjoint functors A → Max(A) and X → C(X)).
A category-theoretic interpretation of the Gelfand-Naimark theorem. The nonunital vesrion of Gelfand's theory (a survey).
- Lecture 11 (10.04.2020) (video)
(lecture notes)
- The continuous
functional calculus in C*-algebras. The spectral mapping theorem and the superposition property for the functional calculus.
The semicontinuity of the spectrum in Banach algebras. The joint continuity of the functional calculus.
- Lecture 12 (17.04.2020) (video)
(lecture notes)
- Positive elements in C*-algebras.
Properties of the set of positive elements. Square roots. Kaplansky's theorem (x*x ≥ 0). Characterizations
of positivity. Sesquilinear forms. Polarization. Hermitian forms and their characterization.
The sesquilinear and the quadratic form associated to an operator between Hilbert spaces.
Characterizations of selfadjoint and positive operators in terms of quadratic forms.
- Lecture 13 (24.04.2020) (video)
(lecture notes)
- Characterizations of isometries,
coisometries, unitary operators, and orthogonal projections in algebraic terms. Partial isometries: several equivalent
definitions. The polar decomposition of bounded linear operators between Hilbert spaces. The uniqueness of the
polar decomposition.
- Lecture 14 (08.05.2020) (video)
(lecture notes)
- *-representations and *-modules.
Cyclic *-modules. Examples. The functional model for a cyclic *-module over C(X). The correspondence between
normal operators and *-modules over C(K). Relations between *-module morphisms and intertwining operators.
*-cyclic operators and their relation to cyclic *-modules. The functional model for a *-cyclic normal
operator.
- Lecture 15 (15.05.2020) (video)
(lecture notes)
- Hilbert direct sums of Hilbert spaces
and of *-modules. A decomposition of a *-module into a Hilbert direct sum of cyclic submodules.
The functional model for a *-module over C(X). The functional model for a normal operator (Spectral Theorem I).
The weak measure topology and the weak operator topology. The separate continuity of the multiplication in
(B(X),WMT) and in (B(H),WOT). The Borel functional calculus for a normal operator (Spectral Theorem II).
- Lecture 16 (22.05.2020) (video)
(lecture notes)
- The correspondence between bounded
linear operators between Hilbert spaces and bounded sesquilinear forms. Extension of representations of C(X) to B(X).
A corollary: the existence of the Borel functional calculus for a normal operator. Spectral measures and associated complex
measures. Examples. Integration of bounded measurable functions with respect to a spectral measure.
The correspondence between finitely additive spectral measures and representations of B_{A}(X).
Regular spectral measures on a compact topological space. The correspondence between regular spectral measures and
WMT-WOT-continuous representations of B(X). A corollary: the spectral decomposition of a normal operator
(Spectral Theorem III). Concluding remarks: multiplicity theory and classification of normal operators.
- 22.05.2020. Individual problem discussion (Zoom, Skype, ...)
- 15.05.2020. Individual problem discussion (Zoom, Skype, ...)
- 08.05.2020. Individual problem discussion (Zoom, Skype, ...)
- 24.04.2020. Individual problem discussion (Zoom, Skype, ...)
- 17.04.2020. Individual problem discussion by Skype
- 10.04.2020. Individual problem discussion by Skype
- 27.03.2020. Individual problem discussion by Skype
- 20.03.2020. Individual problem discussion by Skype
- 06.03.2020. Individual problem discussion
- 28.02.2020. Traditional seminar
- 21.02.2020. Individual problem discussion
- 14.02.2020. Traditional seminar
- 07.02.2020. Individual problem discussion
- 31.01.2020. Traditional seminar
- 24.01.2020. Individual problem discussion (прием задач)
- 17.01.2020. Traditional seminar (problem discussion at the blackboard)
You are allowed to tell your solutions to an instructor
after the deadline as well, but you earn half the points for that.
Exercises marked by "-B" are optional.
If you solve such exercises, you will earn bonus points.