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dc.contributor.advisorNarayanan, E K
dc.contributor.authorRupam, Rishika
dc.date.accessioned2011-08-09T05:01:04Z
dc.date.accessioned2018-07-31T06:08:46Z
dc.date.available2011-08-09T05:01:04Z
dc.date.available2018-07-31T06:08:46Z
dc.date.issued2011-08-09
dc.date.submitted2010
dc.identifier.urihttps://etd.iisc.ac.in/handle/2005/1344
dc.identifier.abstracthttp://etd.iisc.ac.in/static/etd/abstracts/1738/G23708-Abs.pdfen_US
dc.description.abstractIn this thesis, we have dealt primarily with two function algebras. The first one is the space of all holomorphic functions on the unit disc D in the complex plane which are continuous up to the boundary, denoted by A(D). The second one is H1(D), the space of all bounded analytic functions on D. We study results that characterize their maximal ideals. We start with necessary definitions and recall some useful results. In particular, the factorization of Hp functions in terms of Blaschke products, inner and outer functions is stated. Using this factorization, we provide an exposition of a beautiful result, originally by Beurling and rediscovered by Rudin, on the closed ideals of A(D). A maximality theorem by Wermer, which proves that A(D) is itself a maximal closed ideal of H1(D) is proved next. In chapter three, we expand our horizon and look at H1(D) as a dual space to characterize its weak-* closed maximal ideals. In the process we come across the shift operator and a theorem by Beurling, on the shift invariant subspaces of H2(D). We return in our quest to find out more about the maximal ideals of H1(D). The corona theorem states that the maximal ideals of the form Mτ = {ƒ ε H1(D) : ƒ (τ)=0} where τ is in D, are dense in the space of maximal ideals equipped with the Gelfand topology. We describe two approaches to the theorem, one that uses a lemma by Carleson on the existence and special properties of a contour in D. This is followed by a shorter and much more elegant proof by Wolff that uses elementary properties of Hp functions to achieve the same end. We conclude by presenting a proof of the Toeplitz corona theorem.en_US
dc.language.isoen_USen_US
dc.relation.ispartofseriesG23708en_US
dc.subjectAnalytic Functionen_US
dc.subjectUnit Discen_US
dc.subjectBeurling-Rudin Theoremen_US
dc.subjectCorona Theoremen_US
dc.subjectToeplitz Corona Theoremen_US
dc.subject.classificationMathematicsen_US
dc.titleBounded Analytic Functions On The Unit Discen_US
dc.typeThesisen_US
dc.degree.nameMSen_US
dc.degree.levelMastersen_US
dc.degree.disciplineFaculty of Scienceen_US


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