Title
Semi-Fredholm operators on Hilbert C*-modules: doctoral dissertation
Creator
Ivković, Stefan G., 1989-
CONOR:
93646857
Copyright date
2021
Object Links
Select license
Autorstvo-Nekomercijalno-Bez prerade 3.0 Srbija (CC BY-NC-ND 3.0)
License description
Dozvoljavate samo preuzimanje i distribuciju dela, ako/dok se pravilno naznačava ime autora, bez ikakvih promena dela i bez prava komercijalnog korišćenja dela. Ova licenca je najstroža CC licenca. Osnovni opis Licence: http://creativecommons.org/licenses/by-nc-nd/3.0/rs/deed.sr_LATN. Sadržaj ugovora u celini: http://creativecommons.org/licenses/by-nc-nd/3.0/rs/legalcode.sr-Latn
Language
Serbian
Cobiss-ID
Theses Type
Doktorska disertacija
description
Datum odbrane: 07.03.2022.
Other responsibilities
Academic Expertise
Prirodno-matematičke nauke
Academic Title
-
University
Univerzitet u Beogradu
Faculty
Matematički fakultet
Alternative title
Полу-Фредхолмови оператори на Хилбертовим C∗-модулима
Publisher
[S. Ivković]
Format
152 str.
description
Mathematics - Analysis, operator theory and operator algebra / Математика - Анализа, теориjа оператора и алгебре оператора
Abstract (en)
In the first part of the thesis, we establish the semi-Fredholm theory on Hilbert C∗-
modules as a continuation of the Fredholm theory on Hilbert C∗-modules which was introduced
by Mishchenko and Fomenko. Starting from their definition of C∗-Fredholm operator, we give
definition of semi-C∗-Fredholm operator and prove that these operators correspond to one-sided
invertible elements in the Calkin algebra. Also, we give definition of semi-C∗-Weyl operators
and semi-C∗-B-Fredholm operators and obtain in this connection several results generalizing
the counterparts from the classical semi-Fredholm theory on Hilbert spaces. Finally, we consider
closed range operators on Hilbert C∗-modules and give necessary and sufficient conditions for
a composition of two closed range C∗-operators to have closed image. The second part of
the thesis is devoted to the generalized spectral theory of operators on Hilbert C∗-modules.
We introduce generalized spectra in C∗-algebras of C∗-operators and give description of such
spectra of shift operators, unitary, self-adjoint and normal operators on the standard Hilbert C∗-
module. Then we proceed further by studying generalized Fredholm spectra (in C∗-algebras) of
operators on Hilbert C∗-modules induced by various subclasses of semi-C∗-Fredholm operators.
In this setting we obtain generalizations of some of the results from the classical spectral
semi-Fredholm theory such as the results by Zemanek regarding the relationship between the
spectra of an operator and the spectra of its compressions. Also, we study 2×2 upper triangular
operator matrices acting on the direct sum of two standard Hilbert C∗-modules and describe
the relationship between semi-C∗-Fredholmness of these matrices and of their diagonal entries.
Abstract (sr)
У првом делу тезе успостављамо полу-Фредхолмову теориjу на Хилбертовим C∗-
модулима као наставак Фредхолмове теориjе на Хилбертовим C∗-модулима коjу су увели
Мишченко и Фоменко. Полазећи од њихове дефинициjе C∗-Фредхолмових оператора, даjе-
мо дефинициjу полу-C∗-Фредхолмовог оператора и доказуjемо да ти оператори одговараjу
jеднострано инвертибилним елементима у Калкиновоj алгебри. Такође, даjемо дефиници-
jу полу-C∗-Ваjлових оператора и полу-C∗-Б-Фредхолмових оператора и добиjамо с тим
у вези више резултата коjи генерализуjу пандане из класичне полу-Фредхолмове теориjе
на Хилбертовим просторима. На краjу, разматрамо операторе са затвореном сликом на
Хилбертовим C∗-модулима и даjемо потребне и довољне услове да композициjа два C∗-
оператора са затвореном сликом има затворену слику. Други део тезе посвећен jе генера-
лизованоj спектралноj теориjи оператора на Хилбертовим C∗-модулима. За C∗-операторе
дефинишемо генерализоване спектре у C∗-алгебри и даjемо опис таквих спектара у кон-
кретном случаjу оператора помака, унитарних, самоадjонгованих и нормалних оператора
на стандардном Хилбертовом C∗-модулу. Затим настављамо даље проучаваjући генера-
лизоване Фредхолмове спектре (у C∗-алгебрама) оператора на Хилбертовим C∗-модулима
индукованим различитим подкласама полу-C∗-Фредхолмових оператора. У овом контек-
сту добиjамо уопштење неких резултата из класичне спектралне полу-Фредхолмове теори-
jе, као што су Земанекови резултати у вези релациjа између спектара оператора и спектара
њихових компресиjа. Такође, проучавамо 2 × 2 горње триjангуларне операторске матрице
коjе делуjу на директноj суми два стандардна Хилбертова C∗-модула и описуjемо однос
између полу-C∗-Фредхолмности ових матрица и њихових диjагоналних елемената.
Authors Key words
Hilbert C∗-module, semi-C∗-Fredholm operator, semi-C∗-Weyl operator, semi-C∗-
B-Fredholm operator, essential spectrum, Weyl spectrum, perturbation of spectra, compression
Authors Key words
Хилбертов C∗-модул, полу-C∗-Фредхолмов оператор, полу-C∗-Ваjлов опе-
ратор, полу-C∗-Б-Фредхолмов оператор, есенциjални спектар, Ваjлов спектар, пертурба-
циjе спектра, компресиjе
Classification
517.983.24(043.3)
Type
Tekst
Abstract (en)
In the first part of the thesis, we establish the semi-Fredholm theory on Hilbert C∗-
modules as a continuation of the Fredholm theory on Hilbert C∗-modules which was introduced
by Mishchenko and Fomenko. Starting from their definition of C∗-Fredholm operator, we give
definition of semi-C∗-Fredholm operator and prove that these operators correspond to one-sided
invertible elements in the Calkin algebra. Also, we give definition of semi-C∗-Weyl operators
and semi-C∗-B-Fredholm operators and obtain in this connection several results generalizing
the counterparts from the classical semi-Fredholm theory on Hilbert spaces. Finally, we consider
closed range operators on Hilbert C∗-modules and give necessary and sufficient conditions for
a composition of two closed range C∗-operators to have closed image. The second part of
the thesis is devoted to the generalized spectral theory of operators on Hilbert C∗-modules.
We introduce generalized spectra in C∗-algebras of C∗-operators and give description of such
spectra of shift operators, unitary, self-adjoint and normal operators on the standard Hilbert C∗-
module. Then we proceed further by studying generalized Fredholm spectra (in C∗-algebras) of
operators on Hilbert C∗-modules induced by various subclasses of semi-C∗-Fredholm operators.
In this setting we obtain generalizations of some of the results from the classical spectral
semi-Fredholm theory such as the results by Zemanek regarding the relationship between the
spectra of an operator and the spectra of its compressions. Also, we study 2×2 upper triangular
operator matrices acting on the direct sum of two standard Hilbert C∗-modules and describe
the relationship between semi-C∗-Fredholmness of these matrices and of their diagonal entries.
“Data exchange” service offers individual users metadata transfer in several different formats. Citation formats are offered for transfers in texts as for the transfer into internet pages. Citation formats include permanent links that guarantee access to cited sources. For use are commonly structured metadata schemes : Dublin Core xml and ETUB-MS xml, local adaptation of international ETD-MS scheme intended for use in academic documents.