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Operator theory

2021/2022
Programme:
Financial Mathematics, Second cycle
Year:
1 ali 2 year
Semester:
first or second
Kind:
optional
Group:
M1
ECTS:
6
Language:
slovenian, english
Hours per week – 1. or 2. semester:
Lectures
3
Seminar
0
Tutorial
2
Lab
0
Content (Syllabus outline)

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.

Readings

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.

Objectives and competences

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

Intended learning outcomes

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.

Learning and teaching methods

Lectures, exercises, homeworks, consultations

Assessment

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

Lecturer's references

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]