Enrollment into the program.
Positive result from qoloquia (or written exam) is necessary to enter the oral exam.
Introduction to functional analysis
2020/2021
Programme:
Physics, Second Cycle
Orientation:
Mathematical physics
Year:
1. in 2. year
Semester:
first or second
Kind:
optional
Group:
M1
ECTS:
5
Language:
slovenian, english
Course director:
Hours per week – 1. or 2. semester:
Lectures
2
Seminar
0
Tutorial
2
Lab
0
Content (Syllabus outline)
Hilbert space. Orthonormal systems. Fourier series.
Bessel's inequality. Completeness. Parseval's theorem.
Linear operators and functionals on Hilbert spaces.
Representation theorem for linear functionals.
Adjoint operator. Selfadjoint and normal operators.
Projectors and idempotents. Invariant subspaces.
Compact operators. Spectrum of compact operators.
Ries'z decomposition of compact operators.
Diagonalizability of compact selfadjoint operators.
Application: Sturm-Liouville problem.
Readings
J. B. Conway: A Course in Functional Analysis, 2nd edition, Springer, New York, 1990.
D. H. Griffel: Applied Functional Analysis, Dover Publications, Mineola, 2002.
E. Zeidler: Applied Functional Analysis : Main Principles and Their Applications, Springer, New York, 1995.
Objectives and competences
Objectives: Student gets familiar with the basic facts about Hilbert spaces and operators acting on Hilbert spaces. Sturm-Liouville problem is presented in order to see possible applications.
Competences: Knowledge and understanding of basic principles of functional analysis.
Intended learning outcomes
Knowledge and understanding:
Understanding of Hilbert space theory from both theoretical and applied aspects.
Application:
Functional Analysis can be applied in Sciences, Technology, and even in Social Sciences.
Reflection:
Understanding of mathematical theory based on applications.
Transferable skills:
Identification of problems and solving them. The formulation of nonmathematical
problems in the language of mathematics. The use of adequate literature in Slovene and foreign languages.
Learning and teaching methods
Lectures, exercises, homeworks, consultations
Assessment
2 midterm exams or final written exam
grading homeworks, oral exam
grading: 5 (fail), 6-10 (pass) (according to the Statute of UL)
Lecturer's references
Roman Drnovšek:
– DRNOVŠEK, Roman. An irreducible semigroup of idempotents. Studia Mathematica, ISSN 0039-3223, 1997, let. 125, št. 1, str. 97-99 [COBISS-SI-ID 7436633]
– 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. 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]
Bojan Peter Magajna:
– MAGAJNA, Bojan. On completely bounded bimodule maps over W[ast]-algebras. Studia Mathematica, ISSN 0039-3223, 2003, t. 154, fasc. 2, str. 137-164 [COBISS-SI-ID 12278105]
– MAGAJNA, Bojan. Duality and normal parts of operator modules. Journal of functional analysis, ISSN 0022-1236, 2005, vol. 219, no. 2, str. 306-339 [COBISS-SI-ID 13366105]
– MAGAJNA, Bojan. On tensor products of operator modules. Journal of operator theory, ISSN 0379-4024, 2005, vol. 54, no. 2, str. 317-337 [COBISS-SI-ID 13920089]
Peter Šemrl:
– ŠEMRL, Peter, VÄISÄLÄ, Jussi. Nonsurjective nearisometries of Banach spaces. Journal of functional analysis, ISSN 0022-1236, 2003, vol. 198, no. 1, str. 268-278 [COBISS-SI-ID 12371545]
– ŠEMRL, Peter. Generalized symmetry transformations on quaternionic indefinite inner product spaces: an extension of quaternionic version of Wigner's theorem. Communications in Mathematical Physics, ISSN 0010-3616, 2003, vol. 242, no. 3, str. 579-584 [COBISS-SI-ID 12770649]
– ŠEMRL, Peter. Applying projective geometry to transformations on rank one idempotents. Journal of functional analysis, ISSN 0022-1236, 2004, vol. 210, no. , str. 248-257 [COBISS-SI-ID 13012825]
