# What is a Composition Operator

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```					               What is a Composition Operator?

Carl C. Cowen

Bridge to Research Seminar
March 5, 2007

Abstract

Operator Theory is a branch of functional analysis and is an extension of the
study of linear transformations on ﬁnite dimensional vector spaces to an inﬁnite
dimensional setting. Operator Theory emphasizes continuous linear transformations
on complete normed spaces and considers their properties from the perspective of
analysis.
A Hilbert space is a complete normed space in which the norm is derived from an
inner product; a Hilbert space is a generalization of Euclidean space. In this talk, the
underlying spaces will be Hilbert spaces whose vectors are functions that are analytic
on the unit disk in the complex plane.
If H is a Hilbert space of analytic functions on the unit disk and ϕ is an analytic
function mapping the disk into itself, then for f in H, the equation

Cϕ f = f ◦ ϕ

deﬁnes a composition operator on H.
Composition operators form a class of concrete examples which help us
understand properties of general operators, a class that is more complicated than the
“normal” operators (which generalize diagonalizable transformations) and broad
enough to represent a wide variety of operators.
The goal of the study of composition operators is to connect the geometric and
analytic properties of the function ϕ with the operator theoretic properties of the
operator Cϕ .
The talk will present some speciﬁc examples of such connections and will
illustrate the dependence of the properties of the operator on the behavior of ϕ near
its distinguished ﬁxed point in the closed disk.

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