Operator theory

2022/2023
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, ascent and descent of an operator, Riesz decomposition and spectral theory of compact operators
• Invariant subspaces, Schauder's fixed point theorem, Lomonosov's theorem and consequences, triangularizability of compact operators
• Fredholm operators and their index, the index theorem, Atkinson's perturbation theorems, Calkin algebra and essential spectrum, polynomially compact operators
• Banach algebras, Riesz functional calculus, dependence of the spectrum on the algebra, isolated points of the spectrum and Laurent series expansion of the resolvent, essential spectrum and Riesz operators, essential spectral radius
• Additional topic: strictly singular operators, example of a strictly singular non-compact operator, Kato's characterization of strictly singular operators, spectral theory of strictly singular operators
• Additional topic: Schauder basis of a Banach space and basic sequences, examples of spaces with Schauder basis, classical sequence spaces, approximation property, density of finite-rank operators in compact operators

Readings

• Y. A. Abramovich, C. D. Aliprantis: An invitation to operator theory. Graduate Studies in Mathematics 50, American Mathematical Society, Providence, RI, 2002.
• Y. A. Abramovich, C. D. Aliprantis: Problems in operator theory. Graduate Studies in Mathematics 51, American Mathematical Society, Providence, RI, 2002.
• F. Albiac, N. J. Kalton: Topics in Banach space theory, Graduate Texts in Mathematics 233, Springer, New York, 2006.
• 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.
• H. Radjavi, P. Rosenthal: Simultaneous triangularization, Universitext, Springer-Verlag, New York, 2000.
• 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

Type (examination, oral, coursework, project):
homeworks
written exam
oral exam
Grading: 1-5 (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]
Marko Kandić:
KANDIĆ, Marko. On algebras of polynomially compact operators. Linear and Multilinear Algebra, ISSN 0308-1087, 2016, vol. 64, no. 6, str. 1185-1196. [COBISS-SI-ID 17493337]
KANDIĆ, Marko. Ideal-triangularizability of nil-algebras generated by positive operators. Proceedings of the American Mathematical Society, ISSN 0002-9939, 2011, vol. 139, no. 2, str. 485-490. [COBISS-SI-ID 15710809]
DRNOVŠEK, Roman, KANDIĆ, Marko. Positive operators as commutators of positive operators. Studia Mathematica, ISSN 0039-3223, 2019, tom 245, str. 185-200. [COBISS-SI-ID 18407769]