Compact operators on Banach spaces.The Schauder fixed point theorem.

Invariant subspaces. Lomonosov’s theorem. The Riesz decomposition of a compact operator.Fredholm operators. The Calkin algebra. The essential spectrum.Partial isometries and unitary operators.The Schmidt representation of a compact operator.Hilbert-Schmidt operators. Duality between the algebra of all bounded operators, the algebra of all trace-class operators and the algebra of all compact operators.The spectrum of normal operators.The spectral theorem for normal operators (in the multiplication operator form and in the integral form).

The Fuglede-Putnam theorem.

# Operator theory

R. Bhatia: Notes on Functional Analysis, Texts and Readings in Mathematics 50, Hindustan Book Agency, New Delhi, 2009.

J. B. Conway: A Course in Functional Analysis, 2nd edition, Springer, New York, 1990.

I. Gohberg, S. Goldberg, M. A. Kaashoek: Classes of Linear Operators I, Birkhäuser, Basel, 1990.

G. K. Pedersen: Analysis Now, Springer, New York, 1996.

I. Vidav: Linearni operatorji v Banachovih prostorih, DMFA-založništvo, Ljubljana, 1982.

Treatment of some classes of bounded linear operators on Hilbert and Banach spaces.

Knowledge and understanding: Knowledge of some classes of linear operators, the ability to apply the acquired knowledge.

Application: Operator theory is used in natural sciences and other areas of science such as economics.

Reflection: Understanding of the theory, strengthened by examples.

Transferable skills: Identifying and solving problems. Ability to use a wide range of references.

Lectures, exercises, homeworks, consultations

Homeworks

Written exam

Oral exam

grading: 5 (fail), 6-10 (pass) (according to the Statute of UL)

Roman Drnovšek:

DRNOVŠEK, Roman. Common invariant subspaces for collections of operators. Integral equations and operator theory, ISSN 0378-620X, 2001, vol. 39, no. 3, str. 253-266. [COBISS-SI-ID 10597721]

DRNOVŠEK, Roman. A generalization of Levinger's theorem to positive kernel operators. Glasgow mathematical journal, ISSN 0017-0895, 2003, vol. 45, part 3, str. 545-555. [COBISS-SI-ID 12825945]

DRNOVŠEK, Roman. Invariant subspaces for operator semigroups with commutators of rank at most one. Journal of functional analysis, ISSN 0022-1236, 2009, vol. 256, iss. 12, str. 4187-4196. [COBISS-SI-ID 15167321]

Peter Šemrl:

ŠEMRL, Peter. Similarity preserving linear maps. Journal of operator theory, ISSN 0379-4024, 2008, vol. 60, no. 1, str. 71-83. [COBISS-SI-ID 15079257]

ŠEMRL, Peter. Local automorphisms of standard operator algebras. Journal of mathematical analysis and applications, ISSN 0022-247X. [Print ed.], 2010, vol. 371, iss. 2, str. 403-406. [COBISS-SI-ID 15672665]

ŠEMRL, Peter. Symmetries on bounded observables: a unified approach based on adjacency preserving maps. Integral equations and operator theory, ISSN 0378-620X, 2012, vol. 72, iss. 1, str. 7-66. [COBISS-SI-ID 16568665]