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.