Functional Analysis 2. Spring 2023

Alexei Yu. Pirkovskii (lectures), Marat Z. Rovinsky (exercises)

Syllabus

Lectures

Lecture 1 (13.01.2023) (video)
Topologically injective operators, isometries, topological and isometric isomorphisms. Topologically injective = bounded below. Open operators and coisometries. Characterizations of open operators. Quotients of normed spaces. The universal property of quotients. Corollaries.
Lecture 2 (20.01.2023) (video)
The completeness of quotients. Dual spaces and dual operators (a survey). The canonical embedding of a normed space into the bidual. Reflexive Banach spaces. Annihilators and preannihilators, their basic properties. The double annihilator theorem. Corollaries.
Lecture 3 (27.01.2023) (video)
The duals of subspaces and quotients. Relations between kernels and images of operators and of their duals. A duality between injective operators and operators with dense image. A duality between topologically injective and surjective operators on Banach spaces. The Closed Image Theorem.
Lecture 4 (03.02.2023)
Johnson's lemma on exact sequences of Banach spaces. Fredholm operators. The Fredholm index. Examples. The additivity of the index. Kato's lemma (the image of a Fredholm operator is closed). The fredholmness and the index of the dual operator. A survey of compact operators: definition, simplest examples and counterexamples, properties of the set of compact operators, approximation of Hilbert-space-valued compact operators by finite rank operators. Schauder's theorem on the compactness of the dual operator.
Lecture 5 (10.02.2023) (video)
The Riesz-Schauder theory: Riesz's theorem on operators "1+compact", Fredholm alternative, abstract Fredholm theorems in Schauder's form. Topological direct sums and their characterization in terms of projections.
Lecture 6 (17.02.2023)
Complemented subspaces of normed spaces. Characterizations of complemented subspaces. Finite-dimensional subspaces and closed subspaces of finite codimension are complemented. The Nikolskii–Atkinson criterion for Fredholm operators. Banach algebras. Examples.
Lecture 7 (03.03.2023) (video)
The spectrum of an algebra element. Examples. The behavior of the spectrum under homomorphisms. Spectrally invariant subalgebras. The polynomial spectral mapping theorem. Properties of the multiplicative group of a Banach algebra. The automatic continuity of characters (i.e., of C-valued homomorphisms) of a Banach algebra. The compactness and the nonemptiness of the spectrum. The Gelfand-Mazur theorem. The spectral radius of a Banach algebra element.
Lecture 8 (10.03.2023) (video)
The Calkin algebra. The essential spectrum of a linear operator. The stability of the index under "small" perturbations. The stability of the index under compact perturbations. Nikolskii's characterization of Fredholm operators of index zero. Toeplitz operators on the Hardy space. The index formula for Toeplitz operators.
Lecture 9 (17.03.2023)
Topological vector spaces. The topology generated by a family of seminorms. Locally convex spaces and their "polynormability". Examples of locally convex spaces (spaces of continuous and smooth functions, the Schwartz space). The continuity criterion for a seminorm on a locally convex space.
Lecture 10 (31.03.2023)
The continuity criterion for a linear operator between locally convex spaces. The domination relation for families of seminorms. Equivalent families of seminorms. Examples. Quotients of locally convex spaces. Linear functionals on locally convex spaces (extension from subspaces, separation of points, separation of points and closed vector subspaces).
Lecture 11 (07.04.2023) (video)
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 for 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. Annihilators (in the setting of dual pairs). Basic properties of annihilators. The double annihilator theorem (for dual pairs and for locally convex spaces). Corollaries: the closure of a vector subspace equals the weak closure; a density criterion for a vector subspace; relations between kernels and images of a linear operator and of the dual operator.
Lecture 12 (14.04.2023)
Equicontinuous families of linear operators. The Banach-Alaoglu-Bourbaki theorem. 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.
Lecture 13 (21.04.2023) (video)
The maximal spectrum and the Gelfand transform for subalgebras of C(X). The adjoint of an operator on a Hilbert space. Properties of the operation of taking the adjoint. Banach ∗-algebras and C*-algebras. Examples. The spectral radius of a normal element.
Lecture 14 (28.04.2023)
The spectral radius of a normal element of a C*-algebra. 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. A category-theoretic interpretation of the Gelfand-Naimark theorem. The continuous functional calculus in C*-algebras.
Lecture 15 (12.05.2023) (video)
The spectral mapping theorem and the superposition property for the functional calculus. Complex measures and integration. Thie Riesz-Markov-Kakutani theorem (statement only). ∗-representations and ∗-modules. The correspondence between normal operators and ∗-modules over C(K). Relations between ∗-module morphisms and intertwining operators. Cyclic ∗-modules. Examples.
Lecture 16 (19.05.2023) (video)
The functional model for a cyclic ∗-module over C(X). ∗-cyclic operators and their relation to cyclic ∗-modules. The functional model for a ∗-cyclic normal operator. Hilbert direct sums of Hilbert spaces and of ∗-modules. Decomposing ∗-modules into Hilbert direct sums of cyclic submodules. The functional model for a ∗-module over C(X). The functional model for a normal operator (Spectral Theorem). A survey of other versions of the Spectral Theorem: the Borel functional calculus, spectral decompositions, measurable Hilbert bundles.

Seminars

Exercise sheets

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.

Midterm program (24.03.2023)

Exam program (26.05.2023 and 02.06.2023)

Contacts and availability of the instructors and TA's

Grading rules

References