Operator Theory

Objectives

At the end of this course students will have acquired basic knowledge and skills in the area of operator theory in order to:

 - Understand advanced contents in the area;

- Being able to start research on a topic in the area.

General characterization

Code

11639

Credits

6.0

Responsible teacher

Cláudio António Raínha Aires Fernandes

Hours

Weekly - Available soon

Total - Available soon

Teaching language

Português

Prerequisites

Available soon

Bibliography

A. Böttcher, B. Silbermann, Analysis of Toeplitz operators, Springer, 2006.

R. Douglas, Banach algebras techniques in operator theory, Springer, 1998.

I.Gohberg, S. Goldberg, R. Kaashoek, Basic classes of linear operators, Birkhäuser, 2003.

I. Gohberg, N. Krupnik, One-dimensional linear singular integral operators, vol. 1, Birkhäuser, 1992.

C. Murphy, C*-algebras and operator theory, Academic Press, 1990.

S. Roch, P.A. Santos, B. Silbermann, Non-commutative Gelfand theories, Springer, 2011.

Teaching method

Students will be monitored weekly with autonomous study.

Evaluation method

Students will be evaluated according to work reports, oral presentations and work meetings held with the teacher.

Subject matter

1. Linear operators on a Banach space. Closed linear operators. Complemented subspaces and projections. Compact operators. One-sided invertible operators.

2. Fredholm operators. Normally solvable operators. Regularization. Index and trace. Perturbations small in norm. Compact perturbations.

3. Banach algebras. Invertibility and spectrum. Maximal ideals and representations. Some examples.

4. Local principles. Gelfand theory. Allan’s local principle. Norm-preserving localization. Gohberg-Krupnik’s local principle. Simonenko’s local principle. PI-algebras and QI-algebras.

5. Banach algebras generated by idempotents. Algebras generated by two idempotents. An N-idempotents theorem. Algebras with flip.

Programs

Programs where the course is taught: