Hilbert-Schmidtovi operatori i operatori s tragom

master's thesis

• Author: Antonio Tolić
• Mentor: Damir Bakić (mentor)
• Committee member: Damir Bakić (predsjednik povjerenstva)
• Committee member: Igor Pažanin (član povjerenstva)
• Committee member: Ljiljana Arambašić (član povjerenstva)
• Committee member: Marcela Hanzer (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2017-09-26, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Cilj ovog rada bio je upoznati se s Hilbert-Schmidtovim operatorima te operatorima s tragom koje nerijetko nazivamo i nuklearnim operatorima. Hilbert-Schmidtovi operatori i operatori s tragom su pod klase kompaktnih operatora i imaju veliku ulogu u raznim područjima, na primjer u kvantnoj fizici, gdje ih štoviše vrlo često možemo susresti. Uvodno smo naveli osnovne definicije linearne algebre, normiranih prostora i operatora na normiranim prostorima, pri čemu smo naglasak stavili na trag operatora, trenutno na konačnodimenzionalnom prostoru. U sljedećem poglavlju smo ukratko ponovili definicije Banachove algebre koju shvaćamo kao analogon Banachovog prostora. U trećem poglavlju smo promatrali kompaktne operatore. S viškom pozornosti, izrekli smo nužne i bitne definicije, jer ipak kompaktni operatori predstavljaju pravi nadskup Hilbert-Schmidtovih operatora i operatora s tragom. Posebnu smo pažnju dali kompaktnim operatorima na Hilbertovim prostorima. Sljedeće poglavlje koje je bilo na redu su Hilbert-Schmidtovi operatori, dakle uz operatore s tragom, glavna tema ovoga rada. Dali smo osnovnu definiciju Hilbert-Schmidtovih operatora i pokazali da navedena definicija ne ovisi o bazi prostora na kojemu se definiraju. Zatim smo potvrdili da je svaki Hilbert-Schmidtov operator kompaktan. Definirali smo i prostor Hilbert-Schmidtovih operatora te izrekli najbitnija svojstva koja ti operatori posjeduju. U zadnjem poglavlju smo pričali o operatorima s tragom, to jest o nuklearnim operatorima. Definirali smo operator s tragom, pri čemu smo pokazali da ne možemo direktno generalizirati trag konačnodimenzionalnog prostora na beskonačnodimenzionalni prostor. Također smo definirali i prostor operatora s tragom, nakon kojeg smo, analogno Hilbert-Schmidtovim operatorima, iznijeli najbitnija svojstva koja operatori s konačnim tragom posjeduju. Zatim smo analizirali dualne prostore, a za sami kraj smo naveli neka poopćenja nuklearnih operatora.

Abstract (english)

The main purpose of this paper was to explain Hilbert-Schmidt operators and trace-class operators which are often called nuclear operators. Hilbert-Schmidt operators and trace-class operators are subclass of compact operators and they can have many applications in different science areas, for example in quantum physics where they are quite common. In the introductory part we had basic definitions of linear algebra, normed spaces and operators on normed spaces where the focus was on trace-class operators, currently in the finite dimensional space. In the next chapter we have repeated the definitions of Banach algebra which is analogue of Banach space. In third chapter we observed compact operators. With extra regard, the necessary and essential definitions were made, since compact operators are the true superset of Hilbert-Schmidt operators and trace-class operators. Special attention was given to compact operators on Hilbert’s space. The next chapter that was observed are Hilbert-Schmidt operators, which were the main topic beside trace-class operators of this thesis. The basic definition of Hilbert-Schmidt operators was given and showed that definition does not depend on the base of the space in which they are defined. Next we showed that every Hilbert-Schmidt operator is compact. We also defined space of Hilbert-Schmidt operators and said the most important properties of those operators. In the final chapter we talked about trace-class operators or nuclear operators. We defined trace-class operators and showed that they cannot be directly generalized by the trace of the finite dimensional space to the infinite dimensional space. Furthermore, space of trace operators were defined where we, analogue to Hilbert-Schmidt operators, said the most important properties of those operators. In the end we analyzed dual vector space’s, and talked about general nuclear operators.

• Keywords: Hilbert-Schmidtovi operatori; operatori s tragom; nuklearni operatori; konačnodimenzionalni prostor; Banachova algebra; Banachov prostor
• Keywords (english): Hilbert-Schmidt operators; trace-class operators; nuclear operators; finite dimensional space; Banach algebra; Banach space
• Language: croatian
• URN:NBN: urn:nbn:hr:217:019166
• Study programme: Title: Applied Mathematics; Study programme type: university; Study level: graduate; Academic / professional title: magistar/magistra matematike (magistar/magistra matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Access restricted to students and staff of home institution
• Created on: 2018-05-04 09:24:07