# MA4K3 Complex Function Theory

**Lecturer:** Polina Vytnova

**Term(s):** Term 1

**Status for Mathematics students:** List C

**Commitment:** 30 one hour lectures

**Assessment:** Assignments 15%, 3 hour written exam 85%

**Prerequisites:**

MA3B8 Complex Analysis is essential.

MA359 Measure Theory and MA3G7 Functional Analysis I are desirable but not essential.

**Leads To:** PhD level research in function spaces.

**Content:
**1. Problems on the Hardy space.

1.1. Overview. Understanding functions through problems in Complex Function Theory.

1.2. Review of Complex Analysis I, Functional Analysis I and Measure Theory.

1.3. The Hardy space. Basic properties. Other important spaces.

2. Problems on functions.

2.1. Evaluation at one point. Reproducing kernel.

2.2. Multipliers. Bounded functions.

2.3. Existence of boundary values.

2.4. Zero sets. Blaschke products.

2.5. Inner-outer factorization.

3. Problems on operators and functionals.

3.1. Examples of operators. Boundedness. Spectrum. Spectral theorem.

3.2. The shift operator. Subspaces. Polynomials. Cyclicity. Invariant subspaces.

3.3. The restriction operator. Interpolation and sampling. Embeddings.

3.4. Optimization of functionals. Distances. Extremal problems. Cyclicity revisited.

4. What else?

4.1. More spaces and operators. Domains. Several variables. Meromorphic and entire functions. Dirichlet series. Banach spaces. Random functions. More operators.

4.2. More problems. Approximation. Corona. Growth. Other operator properties. Univalence. Completeness. Conformal representations.

**Aims:
**To provide to the students a variety of roads they can follow on their private further research.

To introduce them to the results in analytic function spaces through a fundamental example.

To show to the students how natural problems motivate this study.

**Objectives:
**By the end of the module the student should be able to:

Understand the fundamental properties of the Hardy space.

Understand the fundamental properties of the Hardy space, that this is the case for complex function theory.

Produce proofs of simple facts and solve particular cases of the classical problems.

**Books:
**P. L. Duren, Theory of Hp spaces.

J. E. Garnett, Bounded Analytic Functions.