# Littlewood’s theorem for composition operators on Hardy space

Speaker: Vikramjeet Singh Chandel

Date: 7th April, 2015 (Tuesday)

Time: 09:15 pm – 10:15 pm

Venue: Lecture Hall I, Department of Mathematics

Abstract:

Consider the Hardy space $H^2$ on the unit disc $\mathbb{D}$ of square-summable power series coefficients. To each holomorphic function $\varphi$ that takes $\mathbb{D}$ into itself we associate composition operator $C_{\varphi}$ defined by $C_{\varphi}f = f \circ \varphi$, where $f \in H^2.$ I’ll first present Littlewood’s famous Subordination Principle which establishes that $C_{\varphi}$ maps $H^2$ into itself. Based on this is Littlewood’s Theorem which essentially says that the composition operator is bounded. Further investigations into properties like compactness, spectra of composition operators lead to classical results in complex dynamics like the Denjoy-Wolff Iteration Theorem, Konig’s Linearization Theorem and the Koebe Distortion Theorem.

Area: Complex Analysis, Functional Analysis, Operator Theory