| Abstract | David Hilbert i Erhard Schmidt početkom 20. stoljeća, odnosno 1907. godine, dali su prvu studiju o Hilbert-Schmidtovim operatorima te su po njima ti operatori i dobili ime. Za teoriju Hilbert-Schmidtovih operatora vrlo značajan je John von Neumann, iako se većina njegovih rezultata pripisuje John Wilson Calkinu. U ovom diplomskom radu dan je pregled teorije Hilbert-Schmidtovih operatora koji s nuklearnim operatorima čine istaknutu klasu kompaktnih operatora. Na samom početku dali smo osnovnu definiciju Hilbert-Schmitovih operatora na Hilbertovom prostoru, koju smo kasnije proširili do definicije integralnih Hilbert-Schmidtovih operatora na prostoru pozitivne mjere. Pokazali smo da definicija Hilbert-Schmitovih operatora ovisi samo o prostoru na kojemu se definiraju, ali ne i o bazi tog prostora. Dali smo definiciju Hilbert-Schmidtove norme te pokazali da je skup Hilbert-Schmidtovih operatora zajedno s Hilbert-Schmidtovom normom Banachova algebra te obostrani ideal. Nakon sto smo definirali integralne Hilbert-Schmitove operatore pokazali smo da je svaki Hilbert-Schmitov operator kompaktan operator. Vrlo zanimljiva je teorija o tragu Hilbert-Schmitovih operatora, no prije nego li smo krenuli s tim dijelom teorije morali smo iskazati dvije značajne nejednakosti koje su nam prijeko potrebne za ovaj dio teorije, a to su Hadamardova i Carlemanova nejednakost. Hilbert-Schmidtovi operatori su specifični po svome tragu; za razliku od nuklearnih operatora koji imaju konvergentan trag, trag Hilbert-Schmidtovih operatora definira se za par dvaju operatora. Za funkciju traga Hilbert-Schmidtovih operatora pokazali smo da je to simetrična bilinearna funkcija definirana na produktu skupa Hilbert-Schmidtovih operatora sa samim sobom. Također smo pokazali da je trag dvaju kvazi-nilpotentnih Hilbert-Schmidtovih operatora jednak nuli. Osim traga Hilbert-Schmidtovih operatora, ono što je vrlo zanmljivo u ovom radu je odnos Hilbert-Schmidtovog operatora i analitičke funkcije na okolini njegovog spektra. Pokazali smo da za Hilbert-Schmidtov operator \(A\) i funkciju \(f\) koja je analitička na okolini njegovog spektra vrijedi da je \(f(A)\) Hilbert-Schmidtov operator, a zatim smo pokazali zanimljiv odnos traga ovako definiranih Hilbert-Schmidtovih operatora s jedne strane i vrijednosti analitičkih funkcija u točkama spektra s druge strane. |
| Abstract (english) | The first study regarding Hilbert-Schmidt operators was given in the early 20th century by David Hilbert and Erhard Schmidt, after whom the operators were also named. This thesis presents a theoretical overview of Hilbert-Schmidt operators, which together with nuclear operators form a distinct class of compact operators. At the beginning of the thesis a basic definition of Hilbert-Schmidt operators on a Hilbert space was given. Afterwards, that definition was extended to define Hilbert-Schmidt integral operators on a positive measure space. Furthermore, the definition of Hilbert-Schmidt operators was shown to depend only on the space it was defined on, and not on the choice of the base of that space. The Hilbert-Schmidt norm was defined, the set of all Hilbert-Schmidt operators was proved to be a Banach algebra, as well as a two-sided ideal. After defining Hilbert-Schmidt integral operators, each Hilbert-Schmidt operator was shown to be compact. The theory regarding the trace of Hilbert-Schmidt operators yielded some very interesting results, however before going into the details of that theory, it was necessary to define and prove Hadamard’s and Carleman’s inequalities. Hilbert-Schmidt operators have a distinct trace; unlike nuclear operators which have a convergent trace, their trace is defined for a pair of operators. The trace function of Hilbert-Schmidt operators was shown to be a symmetric bilinear function defined on the product of the set of Hilbert-Schmidt operators with itself. Furthermore, the trace of two quasi-nilpotent Hilbert-Schmidt operators was proved to be zero. Besides the results regarding the trace of Hilbert-Schmidt operators, an interesting relation between a Hilbert-Schmidt operator and a function which is analytic in a neighborhood of the operators spectrum was also established. Given an operator \(A\), and a function \(f\) , analytic in a neighborhood of the operators spectrum, \(f(A)\) was shown to be a Hilbert-Schmidt operator. Finally, an interesting relation between the trace of said Hilbert-Schmidt operators on one hand and the values of analytic functions in the points of the spectrum was also established. |
