Operator Theory
Code
11639
Academic unit
Faculdade de Ciências e Tecnologia
Department
Departamento de Matemática
Credits
6.0
Teacher in charge
Oleksiy Karlovych
Teaching language
Português
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.
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.
6. Toeplitz operators with continuous and piecewise continuous symbols. Fredholm criteria. Banach algebras of Toeplitz operators.
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.