Functional Analysis 2 (Operator Theory). Spring 2022

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

Syllabus

Lectures

Lecture 1 (14.01.2022) (video1, video2)
A survey of compact operators: definition, simplest examples and counterexamples, properties of the set of compact operators, Schauder's theorem on the compactness of the dual operator, approximation of Hilbert-space-valued compact operators by finite rank operators. Some results from the duality theory for Banach spaces: annihilators and preannihilators, the double annihilator theorem, the density criterion for a vector subspace. 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.
Lecture 2 (21.01.2022) (video) (lecture notes)
The Closed Image Theorem. Johnson's lemma on exact sequences of Banach spaces. Fredholm operators. The Fredholm index. Examples. The additivity of the index.
Lecture 3 (28.01.2022) (video) (lecture notes)
Kato's lemma (the image of a Fredholm operator is closed). The fredholmness and the index of the dual operator. The Riesz-Schauder theory: Riesz's theorem on operators "1+compact", Fredholm alternative, abstract Fredholm theorems in Schauder's form.
Lecture 4 (04.02.2022) (video)
Properties of the spectrum of a compact operator. The adjoint of an operator between Hilbert spaces. Properties of the operation of taking the adjoint. The C*-identity. Selfadjoint and normal operators. The spectrum of a selfadjoint operator is real. The spectral radius of a normal operator is equal to the norm. A relation between invariant subspaces of an operator and of its adjoint. The Hilbert–Schmidt theorem (statement).
Lecture 5 (11.02.2022) (video)
The Hilbert–Schmidt theorem (proof). Topological direct sums and complemented subspaces of normed spaces. Characterizations of complemented subspaces. Finite-dimensional subspaces and closed subspaces of finite codimension are complemented.
Lecture 6 (18.02.2022) (video)
The Nikolskii–Atkinson criterion for Fredholm operators. 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.
Lecture 7 (25.02.2022) (video)
The index formula for Toeplitz operators. Topological vector spaces. The topology generated by a family of seminorms. Locally convex spaces and their "polynormability". Examples of locally convex spaces.
Lecture 8 (04.03.2022) (video)
Further examples of locally convex spaces (spaces of smooth functions, the Schwartz space). The continuity criterion for a seminorm on a locally convex space. The continuity criterion for a linear operator between locally convex spaces. The domination relation for families of seminorms. Equivalent families of seminorms. Examples. Linear functionals on locally convex spaces (extension from subspaces, separation of points).
Lecture 9 (11.03.2022) (video)
Quotients of locally convex spaces. Separation of points and closed vector subspaces by linear functionals. 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 10 (25.03.2022) (video)
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. Equicontinuous families of linear operators. The Banach-Alaoglu-Bourbaki theorem. Polars and their basic properties. The bipolar theorem (statement only). Corollaries: Goldstine's theorem, a reflexivity criterion in terms of weak compactness.
Lecture 11 (01.04.2022) (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 12 (08.04.2022) (video)
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. A category-theoretic interpretation of the Gelfand-Naimark theorem. Remarks on the nonunital version of Gelfand's theory.
Lecture 13 (15.04.2022) (video)
The continuous functional calculus in C*-algebras. The spectral mapping theorem and the superposition property for the functional calculus. *-representations and *-modules. Cyclic *-modules. Examples.
Lecture 14 (22.04.2022) (video)
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. 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 I).
Lecture 15 (29.04.2022) (video)
The correspondence between linear operators and sesquilinear forms. 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 extension theorem for representations of C(X) to B(X). The Borel functional calculus for a normal operator (Spectral Theorem II).
Lecture 16 (13.05.2022) (video)
Orthogonal projections. 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 BA(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.

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 (18.03.2022)

Exam program (20.05.2022)

Contacts and availability of the instructors and TA's

Grading rules

References