diplomsko delo
Abstract
V tem diplomskem delu je predstavljena osnovna teorija sebi-adjungiranih
omejenih linearnih operatorjev na Hilbertovem prostoru. V začetnem delu
so zajeti predvsem pojmi in izreki povezani z normiranimi, metričnimi in Banachovimi
prostori. Nato so predstavljeni prostori s skalarnim produktom oz.
Hilbertovi prostori, na katerih je več poudarka. Opisani so pojmi, povezani
z ortogonalnostjo in vpeljani so adjungirani operatorji. Kasneje so obravnavani
sebi-adjungirani omejeni linearni operatorji na Hilbertovih prostorih kot
posebej pomembni operatorji na tem področju. Navedene so različne vrste teh
operatorjev in njihove lastnosti, pomembne za dokaz glavnega izreka v zadnjem
poglavju diplomskega dela. Spektralni izrek za sebi-adjungirane omejene
linearne operatorje je pomembno orodje v funkcionalni analizi, s katerim lahko
vprašanja o sebi-adjungiranih omejenih linearnih operatorjih reduciramo na
vprašanja o ortogonalnih projektorjih. Na njih pa je pogosto lažje odgovoriti.
Keywords
matematika;Hilbertovi prostori;spektralna teorija;normirani prostori;Banachov prostor;linearni operator;spektralni izrek;diplomska dela;
Data
Language: |
Slovenian |
Year of publishing: |
2009 |
Source: |
Maribor |
Typology: |
2.11 - Undergraduate Thesis |
Organization: |
UM FNM - Faculty of Natural Sciences and Mathematics |
Publisher: |
[B. Ferčec] |
UDC: |
51(043.2) |
COBISS: |
16946184
|
Views: |
2344 |
Downloads: |
314 |
Average score: |
0 (0 votes) |
Metadata: |
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Other data
Secondary language: |
English |
Secondary title: |
SPECTRAL THEORY IN HILBERT SPACES |
Secondary abstract: |
The theory of self-adjoint bounded linear operators on Hilbert spaces is presented.
At the beginning we survey some basic facts concerning normed, metric
and Banach spaces. Then we consider Hilbert spaces, i.e. Banach spaces with
an inner product, which are the central topic of this diploma thesis. The
notions related to orthogonality are examined, and adjoint operators are introduced.
Further, the important class of self-adjoint operators is studied in
greater detail. Some special subclasses are considered, and several theorems
needed for the proof of the spectral theorem in the last section are established.
The spectral theorem for self-adjoint operators is an important tool in
functional analysis. It reduces certain questions on such operators to similar
questions on projections, which are considerably easier to handle. |
Secondary keywords: |
Normed space;Banach space;Hilbert space;self-adjoint bounded linear operator. spectral theorem.; |
URN: |
URN:SI:UM: |
Type (COBISS): |
Undergraduate thesis |
Thesis comment: |
Univ. v Mariboru, Fak. za naravoslovje in matematiko, Oddelek za matematiko in računalništvo |
Pages: |
67 f. |
Keywords (UDC): |
mathematics;natural sciences;naravoslovne vede;matematika;mathematics;matematika; |
ID: |
17844 |