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Bounded Analytic Functions On The Unit Disc

Electronic Theses of Indian Institute of Science

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Title Bounded Analytic Functions On The Unit Disc
 
Creator Rupam, Rishika
 
Subject Analytic Function
Unit Disc
Beurling-Rudin Theorem
Corona Theorem
Toeplitz Corona Theorem
Mathematics
 
Description In 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.
 
Contributor Narayanan, E K
 
Date 2011-08-09T05:01:04Z
2011-08-09T05:01:04Z
2011-08-09
2010-03
 
Type Thesis
 
Identifier http://etd.iisc.ernet.in/handle/2005/1344
http://etd.ncsi.iisc.ernet.in/abstracts/1738/G23708-Abs.pdf
 
Language en_US
 
Relation G23708