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master's thesis
Strukturna teorija za normalne operatore
2020. urn:nbn:hr:126:278311
Mercvajler, Karla
Josip Juraj Strossmayer University of Osijek
Department of Mathematics
Chair of Pure Mathematics
Algebra and Calculus Research Group

Cite this document

Mercvajler, K. (2020). Strukturna teorija za normalne operatore (Master's thesis). Retrieved from https://urn.nsk.hr/urn:nbn:hr:126:278311

Mercvajler, Karla. "Strukturna teorija za normalne operatore." Master's thesis, Josip Juraj Strossmayer University of Osijek, Department of Mathematics, 2020. https://urn.nsk.hr/urn:nbn:hr:126:278311

Mercvajler, Karla. "Strukturna teorija za normalne operatore." Master's thesis, Josip Juraj Strossmayer University of Osijek, Department of Mathematics, 2020. https://urn.nsk.hr/urn:nbn:hr:126:278311

Mercvajler, K. (2020). 'Strukturna teorija za normalne operatore', Master's thesis, Josip Juraj Strossmayer University of Osijek, Department of Mathematics, accessed 14 June 2021, https://urn.nsk.hr/urn:nbn:hr:126:278311

Mercvajler K. Strukturna teorija za normalne operatore [Master's thesis]. Osijek: Josip Juraj Strossmayer University of Osijek, Department of Mathematics; 2020 [cited 2021 June 14] Available at: https://urn.nsk.hr/urn:nbn:hr:126:278311

K. Mercvajler, "Strukturna teorija za normalne operatore", Master's thesis, Josip Juraj Strossmayer University of Osijek, Department of Mathematics, Osijek, 2020. Available at: https://urn.nsk.hr/urn:nbn:hr:126:278311

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Metadata
TitleStrukturna teorija za normalne operatore
Title (english)The Structural Theory for Normal Operators
AuthorKarla Mercvajler
MentorIvan Matić (mentor)
Committee memberSuzana Miodragović (predsjednik povjerenstva)
Committee memberIvan Matić (član povjerenstva)
Committee memberMarija Miloloža Pandur (član povjerenstva)
GranterJosip Juraj Strossmayer University of Osijek
Department of Mathematics
(Chair of Pure Mathematics)
(Algebra and Calculus Research Group)
Osijek
Defense date and country2020-10-12, Croatia
Scientific / art field,
discipline and subdiscipline		NATURAL SCIENCES
Mathematics
Algebra
Abstract
U ovom radu se bavimo pitanjem kada je operator na unitarnom prostoru V dijagonalizabilan. Prvo uvodimo pojam adjungiranog operatora te to koristimo da bi definirali normalne operatore, a to su oni za koje operator i njegov adjungirani operator komutiraju. Zatim dokazujemo Strukturni teorem za normalne operatore koji tvrdi da su normalni operatori unitarno dijagonalizabilni s obzirom na ortonormiranu bazu i formuliramo njegov matrični zapis. Nadalje, definiramo ortogonalnu projekciju i dokazujemo njenu karakterizaciju. Pokazali smo da je dekompozicija jedinice ortogonalna ako su svi projektori ortogonalni te to koristimo da definiramo spektralnu dekompoziciju normalnog operatora. U zadnjem dijelu dolazimo do glavnog rezultata ovog rada, a to je Spektralni teorem.
Abstract (english)
In this thesis we deal with the issue when an operator on an inner product space V is diagonalizable. We first introduce the notion of the adjoint of an operator and use this to define normal operators, which are those for which the operator and its adjoint commute with each other. Then, there is the proof of the Structure theorem for normal operators which states that normal operators are diagonal with respect to an orthonormal basis and we formulate his the matrix version. Furthermore, we define orthogonal projection and prove their characterization. It has been shown that if the projections are themselves orthogonal, then the resolutions of identity is orthogonal. We use this to define spectral resolution of normal operators. In the last part we come the main result of this thesis which is the Spectral theorem.
Keywords
Keywords (english)
Languagecroatian
URN:NBNurn:nbn:hr:126:278311
Study programmeTitle: Mathematics; specializations in: Financial and Statistical Mathematics, Mathematics and Computer Science, Industrial and Applied Mathematics
Course: Financial and Statistical Mathematics
Study programme type: university
Study level: graduate
Academic / professional title: magistar/magistra matematike (magistar/magistra matematike)
Type of resourceText
File originBorn digital
Access conditionsOpen access
Terms of useLicence In copyright icon
Created on2021-01-22 07:43:33