Thesis (MSc (Mathematics))--University of Stellenbosch, 2010. / ENGLISH ABSTRACT: This thesis is a study of a generalisation, due to R. Harte (see [9]), of Fredholm
theory in the context of bounded linear operators on Banach spaces
to a theory in a Banach algebra setting. A bounded linear operator T on a
Banach space X is Fredholm if it has closed range and the dimension of its
kernel as well as the dimension of the quotient space X/T(X) are finite. The
index of a Fredholm operator is the integer dim T−1(0)−dimX/T(X). Weyl
operators are those Fredholm operators of which the index is zero. Browder
operators are Fredholm operators with finite ascent and descent. Harte’s generalisation
is motivated by Atkinson’s theorem, according to which a bounded
linear operator on a Banach space is Fredholm if and only if its coset is invertible
in the Banach algebra L(X) /K(X), where L(X) is the Banach
algebra of bounded linear operators on X and K(X) the two-sided ideal of
compact linear operators in L(X). By Harte’s definition, an element a of a
Banach algebra A is Fredholm relative to a Banach algebra homomorphism
T : A ! B if Ta is invertible in B. Furthermore, an element of the form
a + b where a is invertible in A and b is in the kernel of T is called Weyl
relative to T and if ab = ba as well, the element is called Browder. Harte
consequently introduced spectra corresponding to the sets of Fredholm, Weyl
and Browder elements, respectively. He obtained several interesting inclusion
results of these sets and their spectra as well as some spectral mapping
and inclusion results. We also convey a related result due to Harte which
was obtained by using the exponential spectrum. We show what H. du T.
Mouton and H. Raubenheimer found when they considered two homomorphisms.
They also introduced Ruston and almost Ruston elements which led
to an interesting result related to work by B. Aupetit. Finally, we introduce
the notions of upper and lower semi-regularities – concepts due to V. M¨uller.
M¨uller obtained spectral inclusion results for spectra corresponding to upper
and lower semi-regularities. We could use them to recover certain spectral
mapping and inclusion results obtained earlier in the thesis, and some could
even be improved. / AFRIKAANSE OPSOMMING: Hierdie tesis is ‘n studie van ’n veralgemening deur R. Harte (sien [9]) van
Fredholm-teorie in die konteks van begrensde lineˆere operatore op Banachruimtes
tot ’n teorie in die konteks van Banach-algebras. ’n Begrensde lineˆere
operator T op ’n Banach-ruimte X is Fredholm as sy waardeversameling geslote
is en die dimensie van sy kern, sowel as di´e van die kwosi¨entruimte
X/T(X), eindig is. Die indeks van ’n Fredholm-operator is die heelgetal
dim T−1(0) − dimX/T(X). Weyl-operatore is daardie Fredholm-operatore
waarvan die indeks gelyk is aan nul. Fredholm-operatore met eindige styging
en daling word Browder-operatore genoem. Harte se veralgemening is gemotiveer
deur Atkinson se stelling, waarvolgens ’n begrensde lineˆere operator op
’n Banach-ruimte Fredholm is as en slegs as sy neweklas inverteerbaar is in die
Banach-algebra L(X) /K(X), waar L(X) die Banach-algebra van begrensde
lineˆere operatore op X is en K(X) die twee-sydige ideaal van kompakte
lineˆere operatore in L(X) is. Volgens Harte se definisie is ’n element a van
’n Banach-algebra A Fredholm relatief tot ’n Banach-algebrahomomorfisme
T : A ! B as Ta inverteerbaar is in B. Verder word ’n Weyl-element relatief
tot ’n Banach-algebrahomomorfisme T : A ! B gedefinieer as ’n element
met die vorm a + b, waar a inverteerbaar in A is en b in die kern van T is.
As ab = ba met a en b soos in die definisie van ’n Weyl-element, dan word
die element Browder relatief tot T genoem. Harte het vervolgens spektra
gedefinieer in ooreenstemming met die versamelings van Fredholm-, Weylen
Browder-elemente, onderskeidelik. Hy het heelparty interessante resultate
met betrekking tot insluitings van die verskillende versamelings en hulle
spektra verkry, asook ’n paar spektrale-afbeeldingsresultate en spektraleinsluitingsresultate.
Ons dra ook ’n verwante resultaat te danke aan Harte
oor, wat verkry is deur van die eksponensi¨ele-spektrum gebruik te maak.
Ons wys wat H. du T. Mouton en H. Raubenheimer verkry het deur twee
homomorfismes gelyktydig te beskou. Hulle het ook Ruston- en byna Rustonelemente
gedefinieer, wat tot ’n interessante resultaat, verwant aan werk van
B. Aupetit, gelei het. Ten slotte stel ons nog twee begrippe bekend, naamlik
’n onder-semi-regulariteit en ’n bo-semi-regulariteit – konsepte te danke
aan V. M¨uller. M¨uller het spektrale-insluitingsresultate verkry vir spektra
wat ooreenstem met bo- en onder-semi-regulariteite. Ons kon dit gebruik
om sekere spektrale-afbeeldingsresultate en spektrale-insluitingsresultate wat
vroe¨er in hierdie tesis verkry is, te herwin, en sommige kon selfs verbeter
word.
| Identifer | oai:union.ndltd.org:netd.ac.za/oai:union.ndltd.org:sun/oai:scholar.sun.ac.za:10019.1/4265 |
| Date | 03 1900 |
| Creators | Heymann, Retha |
| Contributors | Mouton, S., University of Stellenbosch. Faculty of Science. Dept. of Mathematical Sciences. |
| Publisher | Stellenbosch : University of Stellenbosch |
| Source Sets | South African National ETD Portal |
| Language | English |
| Detected Language | Unknown |
| Type | Thesis |
| Format | 89 p. |
| Rights | University of Stellenbosch |
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