Thesis (Ph.D.)--Flinders University, School of Informatics and Engineering, Dept. of Mathematics. / Typescript bound. Includes bibliographical references: (leaves 135-140) and index. Also available online.

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.

<p>We investigate the spectrum of certain Banach algebras. Properties</p><p>like generators of maximal ideals and generalized Shilov boundaries are studied. In particular we show that if the ∂-equation has solutions in the algebra of bounded functions or continuous functions up to the boundary of a domain D ⊂⊂ Cⁿ then every maximal ideal over D is generated by the coordinate functions. This implies that the fibres over D in the spectrum are trivial and that the projection on Cn of the n − 1 order generalized Shilov boundary is contained in the boundary of D.</p><p>For a domain D ⊂⊂ Cⁿ where the boundary of the Nebenhülle coincide</p><p>with the smooth strictly pseudoconvex boundary points of D we show that there always exist points p ∈ D such that D has the Gleason property at p.</p><p>If the boundary of an open set U is smooth we show that there exist points in</p><p>U such that the maximal ideals over those points are generated by the coordinate functions.</p><p>An example is given of a Riemann domain, Ω, spread over Cⁿ where the fibers over a point p ∈ Ω consist of m > n elements but the maximal ideal over p is generated by n functions.</p>

We investigate the spectrum of certain Banach algebras. Properties like generators of maximal ideals and generalized Shilov boundaries are studied. In particular we show that if the ∂-equation has solutions in the algebra of bounded functions or continuous functions up to the boundary of a domain D ⊂⊂ Cn then every maximal ideal over D is generated by the coordinate functions. This implies that the fibres over D in the spectrum are trivial and that the projection on Cn of the n − 1 order generalized Shilov boundary is contained in the boundary of D. For a domain D ⊂⊂ Cn where the boundary of the Nebenhülle coincide with the smooth strictly pseudoconvex boundary points of D we show that there always exist points p ∈ D such that D has the Gleason property at p. If the boundary of an open set U is smooth we show that there exist points in U such that the maximal ideals over those points are generated by the coordinate functions. An example is given of a Riemann domain, Ω, spread over Cn where the fibers over a point p ∈ Ω consist of m &gt; n elements but the maximal ideal over p is generated by n functions.

Orientadores: Daniela Mariz Silva Vieira, Jorge Tulio Mujica Ascui / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-19T00:22:09Z (GMT). No. of bitstreams: 1 Nachtigall_Cicero_D.pdf: 625263 bytes, checksum: 7dc27b9956ffbb6e717aa95776b7b8c8 (MD5) Previous issue date: 2011 / Resumo: O resumo, na íntegra, poderá ser visualizado no texto completo da tese digital. / Abstract: The complete abstract is available with the full electronic digital thesis or dissertations. / Doutorado / Matematica / Doutor em Matemática

Indiana University-Purdue University Indianapolis (IUPUI) / The main part of this thesis, Chapter 4, contains results on the commutant of a semigroup of operators defined on the Hardy Space of the disk where the operators have hyperbolic non-automorphic symbols. In particular, we show in Chapter 5 that the commutant of the semigroup of operators is in one-to-one correspondence with a Banach algebra of bounded analytic functions on an open half-plane. This algebra of functions is a subalgebra of the standard Newton space. Chapter 4 extends previous work done on maps with interior fixed point to the case of the symbol of the composition operator having a boundary fixed point.

The axiomatic theory of ` Zelazko defines a variety of general spectra where specified axioms are satisfied. However, there arise a number of spectra, usually defined for a single element of a Banach algebra, that are not covered by the axiomatic theory of ` Zelazko. V. Kordula and V. M¨uller addressed this issue and created the theory of regularities. Their unique idea was to describe the underlying set of elements on which the spectrum is defined. The axioms of a regularity provide important consequences. We prove that the set of Koliha-Drazin invertible elements, which includes the Drazin invertible elements, forms a regularity. The properties of the spectrum corresponding to a regularity are also investigated. / Mathematical Sciences / M. Sc. (Mathematics)

The axiomatic theory of ` Zelazko defines a variety of general spectra where specified axioms are satisfied. However, there arise a number of spectra, usually defined for a single element of a Banach algebra, that are not covered by the axiomatic theory of ` Zelazko. V. Kordula and V. M¨uller addressed this issue and created the theory of regularities. Their unique idea was to describe the underlying set of elements on which the spectrum is defined. The axioms of a regularity provide important consequences. We prove that the set of Koliha-Drazin invertible elements, which includes the Drazin invertible elements, forms a regularity. The properties of the spectrum corresponding to a regularity are also investigated. / Mathematical Sciences / M. Sc. (Mathematics)

Submitted by Leonardo Cavalcante (leo.ocavalcante@gmail.com) on 2018-04-23T20:47:31Z No. of bitstreams: 1 Arquivototal.pdf: 1433584 bytes, checksum: fb8978802ac3b768c50f569bc4124e5e (MD5) / Made available in DSpace on 2018-04-23T20:47:31Z (GMT). No. of bitstreams: 1 Arquivototal.pdf: 1433584 bytes, checksum: fb8978802ac3b768c50f569bc4124e5e (MD5) Previous issue date: 2014-03-20 / Conselho Nacional de Pesquisa e Desenvolvimento Científico e Tecnológico - CNPq / In this dissertation we develop a rst contact with the theory of Banach Algebras and C*-algebras. As usual of a rst contact, we build the Spectral Theory in Banach algebras with unit. We present the characterization theorems of C *-algebras of Gelfand-Naimark and Gelfand-Naimark-Segal, including the GNS construction. Moreover, we prove a theorem which characterizes all complex homomorphisms in the C*-algebra C(X), as point-evaluation homomorphisms. We also present, as a curiosity, a proof of the Fundamental Theorem of Algebra using the Gelfand-Mazur Theorem. As a prerequisite to the Gelfand-Naimark-Segal's characterization of C *-algebras, we further develop, in the background, the theory of the direct sum of any family of Hilbert spaces. . / Nesta dissertação desenvolveremos um primeiro contato com a Teoria de Álgebras de Banach e C*-álgebras. Como tópico de um primeiro contato, construiremos a Teoria Espectral em Álgebras de Banach com unidade. Apresentaremos os Teoremas de Caracterização de C*-álgebras de Gelfand-Naimark, e Gelfand-Naimark-Segal, incluindo a constru c~ao GNS. Al em disso, provamos um teorema que caracteriza todos os homomor smos complexos na C*-álgebra C(X) como sendo homomor smos de avaliação. Apresentaremos também, como curiosidade, uma prova do Teorema Fundamental da Álgebra a partir do Teorema de Gelfand-Mazur. Como um pré requisito a Caracterização de Gelfand-Naimark-Segal de C*-álgebras, desenvolvemos ainda, em segundo plano, a teoria da soma direta de uma familia qualquer de espaços de Hilbert.