Majee, Satyabarta and D, Venku Naidu
(2018)
Composition Operators and Classical Function
Theory.
Masters thesis, Indian Institute of Technology Hyderabad.
Abstract
(1) The section Linear Fractional Prologue is a context, to be consulted as needed , on the basic
properties and classi�cation of linear fractional transformations. Linear fractional maps play a vital
role in my work, both as agents for changing coordinates and transforming settings.
(2) In the section Fourier series, I discuss how to construct a inner product from given Fourier
series , the Dirichlet Kernel and its properties. Then I give the proof of Plancharal theorem and
Parseval's theorem which play a good role through out my project.
(3) This Littlewood's Theorem section is most important part of my work. After developing
some of the basic properties of H2, here we shows that every composition operator acts boundedly
on the Hilbert space. As pointed out above, this is essentially Littlewood's Subordination Principle.
I present Littlewood's original proof  a beautiful argument that is perfectly transparent in its beauty,
but utterly ba�ing in its lack of geometric insight. Much of conclusion can be regarded as an e�ort
to understand the geometric underpinning of this theorem.
(4) Having established that every composition operator is bounded on H2, we turn to the most
natural followup question: "Which composition operators are compact?" The Chapter Compact
ness:Introduction sets out the motivation for this problem. The property of "boundedness" for
composition operators means that each one takes bounded subsets of H2 to bounded subsets. The
question above asks us to specify precisely how much the inducing map � has to compress the unit
disc into itself in order to insure that the operator C� compresses bounded subsets of H2 into relatively
compact ones.
(5) In Chapter Compactness and Univalence we discover that the geometric soul of Littlewood's
Theorem is bound up in the Schwarz Lemma. Armed with this insight, we are able to characterize
the univalently induced compact composition operators, obtaining a compactness criterion that
leads directly to the JuliaCaratheodory Theorem on the angular derivative.
(6) In Chapter The Angular Derivative, I give the idea of the proof of JuliaCaratheodory Theorem
in a way that emphasizes its geometric content, especially its connection with the Schwarz Lemma.
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