Uvod v funkcionalno analizo

Pogojev za vključitev v delo ni.

Banachovi prostori. Primeri. Končnorazsežni normirani prostori. Kvocienti in produkti normiranih prostorov. Linearni operatorji in funkcionali na Banachovih prostorih. Dualni prostor. Hahn-Banachov izrek in posledice. Separacija konveksnih množic. Izrek o odprti preslikavi. Izrek o zaprtem grafu. Princip enakomerne omejenosti. Hilbertovi prostori. Ortonormirani sistemi. Besslova neenakost. Kompletnost. Fouriereve vrste. Parsevalova enakost. Linearni operatorji in funkcionali na Hilbertovih prostorih. Reprezentacija zveznega linearnega funkcionala. Adjungirani operator. Sebiadjungirani in normalni operatorji. Projektorji in idempotenti. Invariantni podprostori. Kompaktni operatorji. Spekter kompaktnega operatorja. Diagonalizacija kompaktnega sebiadjungiranega operatorja.

1. B. Bollobás: Linear analysis : an introductory course, 2nd ed., Cambridge : Cambridge Univ., cop. 1999.
2. J. B. Conway: A course in functional analysis, 2nd ed., New York : Springer, cop. 1990.
3. Y. Eidelman, V. Milman, A. Tsolomitis: Functional analysis : an introduction, Providence : American Mathematical Society, cop. 2004.
4. D. H. Griffel: Applied functional analysis, Chichester : Ellis Horwood ; New York : J. Wiley & Sons, cop. 1981.
5. M. Hladnik: Naloge in primeri iz funkcionalne analize in teorije mere, Ljubljana : Društvo matematikov, fizikov in astronomov SRS, 1985.
6. E. Zeidler: Applied functional analysis : main principles and their applications, New York : Springer, cop. 1995.

Študent spozna osnovne pojme teorije Hilbertovih prostorov in linearnih operatorjev med njimi. Nekoliko spozna tudi teorijo Banachovih prostorov, ki so posplošitev Hilbertovih prostorov.

Znanje in razumevanje: Razumevanje teorije Hilbertovih prostorov s teoretičnega in uporabnega vidika.
Uporaba: Uporaba funkcionalne analize sega tudi v naravoslovje in druga področja znanosti kot na primer ekonomijo.
Refleksija: Razumevanje teorije na podlagi uporabe.
Prenosljive spretnosti – niso vezane le na en predmet: Sposobnost abstraktnega razmišljanja. Spretnost uporabe domače in tuje literature.

predavanja, vaje, domače naloge, konzultacije

Domače naloge
Izpit iz vaj
Ustni izpit
(ocene: 5 (negativno), 6-10 (pozitivno), ob upoštevanju Statuta UL)

Roman Bessonov:

• BESSONOV, Roman V., Szegő condition and scattering for one-dimensional Dirac operators, Constructive approximation. - ISSN 0176-4276 (Vol. 51, iss. 2, Apr. 2020, str. 273-302), COBISS-SI-ID - 218704899
• BESSONOV, Roman V., Wiener-Hopf operators admit triangular factorization. Journal of operator theory. - ISSN 0379-4024 (Vol. 82, iss. 1, 2019, str. 237-249), COBISS-SI-ID - 218617603
• BESSONOV, Roman V.; DENISOV, Sergey A., A spectral Szegő theorem on the real line, Advances in mathematics. - ISSN 0001-8708 (Vol. 359, [article no.] 106851, Jan. 2020, 41 str.), COBISS-SI-ID - 218696707

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]

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]

Igor Klep:

• KLEP, Igor, VINNIKOV, Victor, VOLČIČ, Jurij. Local theory of free noncommutative functions: germs, meromorphic functions, and Hermite interpolation. Transactions of the American Mathematical Society. Aug. 2020, vol. 373, no. 8, str. 5587-5625. ISSN 0002-9947. [COBISS-SI-ID 23631107]
• HELTON, J. William, KLEP, Igor, MCCULLOUGH, Scott. The tracial Hahn-Banach theorem, polar duals, matrix convex sets, and projections of free spectrahedra. Journal of the European Mathematical Society. 2017, vol. 19, iss. 6, str. 1845-1897. ISSN 1435-9855. [COBISS-SI-ID 18057817]
• HELTON, J. William, KLEP, Igor, MCCULLOUGH, Scott, VOLČIČ, Jurij. Bianalytic free maps between spectrahedra and spectraballs. Journal of functional analysis. Jun. 2020, vol. 278, iss. 11, art. 108472 (61 str.). ISSN 0022-1236. [COBISS-SI-ID 15911683]

Aleksey Kostenko:

• KOSTENKO, Aleksey. Hankel Operators and Applications (Lecture Notes: https://users.fmf.uni-lj.si/kostenko/teach/HankelNotes.pdf), 2020; COBISS-SI-ID - 30308867.
• KOSTENKO, Aleksey, Trace Ideals and Applications (Lecture Notes: https://users.fmf.uni-lj.si/kostenko/teach/IdealsNotes.pdf), 2020; COBISS-SI-ID - 30316803.
• KOSTENKO, Aleksey, NICOLUSSI, Noema, Laplacians on Infinite Graphs, Memoirs of the European Mathematical Society, ISBN - 978-3-98547-025-9, 2023, vol. 3, 2023; viii+232 str.