New norm estimate for composition operators
Preeti Kumari, P. Muthukumar, Jaydeb Sarkar
TL;DR
This work proves a new, sharper norm bound for composition operators $C_{\varphi}$ on the Hardy space $H^2({\mathbb D})$, namely $\|C_{\varphi}\| \leq \left\| \frac{\sqrt{1-|\varphi(0)|^2}}{1-\overline{\varphi(0)}\,\varphi} \right\|_{\infty}$, which in turn is bounded by the classical Littlewood bound $\sqrt{\frac{1+|\varphi(0)|}{1-|\varphi(0)|}}$. The authors derive this bound via reproducing kernel Hilbert space techniques, with the key kernel $k_{\varphi}(z,w)=\dfrac{1-\varphi(z)\overline{\varphi(w)}}{1-z\overline{w}}$, and identify equality cases when $\varphi(0)\neq0$ and $\dfrac{\varphi(0)}{|\varphi(0)|}\in\partial(\varphi(\mathbb D))$. They further develop a general RKHS framework to study boundedness and norms of $C_{\varphi}$ between spaces with given kernels, and provide extensive examples showing the new bound is sharper for a broad class of symbols, including some with $\|\varphi\|_{\infty}=1$ where $C_{\varphi}$ is noncompact. The paper also compares the new bound with Schwartz’s estimates, showing the new bound is sharper in many cases and correcting previous claims. Overall, the results deepen understanding of composition operators through kernel methods and yield sharper, verifiable norm bounds with clear equality criteria.
Abstract
The classical Littlewood's theorem establishes boundedness and provides a norm estimate for composition operators on the Hardy space. In this paper, we offer an alternative proof of boundedness and derive a new norm estimate that improves upon the classical bound given by Littlewood's theorem.