| OPOMBE | Bibliografija: str. 76-78 // Univ. Ljubljana, Fakulteta za naravoslovje in tehnologijo, VTO matematika in mehanika // Najbo ▫${\cal H}$▫ Hilbertov prostor in ▫${\cal B(H)}$▫ algebra vseh omejenih linearnih operatorjev na njem. V prvem delu disertacije raziskujemo sistem enačb ▫$$\sum_{j=1}^m X_jA_iY_j = B_i,\quad i=1,...,r$$▫ kjer so ▫$A_i$▫ in▫$B_i$▫ znani, ▫$X_j$▫ in ▫$Y_j$▫ pa neznani elementi algebre ▫${\cal B(H)}$▫. V primeru ▫$m=r$▫ podamo potreben in zadosten pogoj, ki mu morajo zadoščati operatorji ▫$A_i$▫, da bo gornji sistem rešljiv pri poljubnih ▫$B_i \in {\cal B(H)}$▫. Dobljeni rezultat (katerega milejša oblika velja tudi v splošnejših Banachovih algebrah z enim samim maksimalnim idealom) jeuporaben pri proučevanju vprašanja, kdaj je zaloga vrednosti danega elementarnega operatorja vsebovana v kakem idealu algebre ▫${\cal B(H)}$▫. Kot posledico spoznamo, da na Calkinovi algebri ni neničelnih kompaktnih elementarnih operatorjev. Drugi del naloge se omejuje na posebne primere elementarnih operatorjev, predvsem na t.i. posplošena odvajanja. Med drugimpokažemo, kako se izraža bistveni numerični zaklad zožitve posplošenega odvajanja na Hilbert-Schmidtov razred ▫$C^2({\cal H})$▫ z numeričnima in bistvenima numeričnima zakladoma operatorjev, ki ga inducirata. Nadalje opredelimo subnormalna in kvazinormalna posplošena odvajanja na ▫$C^2({\calH})$▫. // Let ▫${\cal H}$▫ be a Hilbert space and ▫${\cal B(H)}$▫ the algebra of all bounded linear operators on ▫${\cal H}$▫. In the first part of the dissertation the system of equations ▫$$\sum_{j=1}^m X_jA_iY_j = B_i, \quad i=1,...,r$$▫ is investigated, where ▫$A_i$▫, ▫$B_i$▫ are known and ▫$X_j$▫, ▫$Y_j$▫ are unknown elements of ▫${\cal B(H)}$▫. In the case ▫$m=r$▫ the necessary and sufficient condition on operators ▫$A_i$▫ is given that the above system is solvable for arbitrary ▫$B_i \in {\cal B(H)}$▫. This result (the weaker form of which holds in fact also in more general algebras with the unique maximal ideal) is supplied to study the guestion, when is the range of an elementary operator contained in an idealof ▫${\cal B(H)}$▫. As a consequence, it follows that there are no non-zerocompact elementary operators on the Calkin algebra. The second part of the dissertation is devoted to the special classes of elementary operators, mainly to the generalized derivations. Among other results, the essential numerical range of the restriction of a generalized derivation to the Hilbert-Schmidt class ▫$C^2({\cal H})$▫ is expressed in terms of the numerical and the essential numerical ranges of the operators that induce the derivation, and the subnormal and quasinormal generalized derivations on ▫$C^2({\cal H})$▫ are characterized.Način dostopa (URN): http://www.dlib.si/?urn=URN:NBN:SI:doc-7S8MBT5J |