SPECTRAL THEORY IN HILBERT SPACES

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.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