Normaloid Weighted Composition Operators on $H^2$

TL;DR

This paper characterizes when weighted composition operators on the Hardy space are normaloid, providing spectral and norm properties, especially focusing on operators with Denjoy-Wolff points inside or on the boundary of the unit disk.

Contribution

It offers an exact characterization of normaloid weighted composition operators on H^2 based on the Denjoy-Wolff point and computes their spectral properties, extending understanding of these operators.

Findings

Characterization of normaloid weighted composition operators with interior Denjoy-Wolff points.

Determination of spectral radius, essential spectral radius, and essential norm for certain non-compact operators.

Sufficient conditions for normaloid operators with boundary Denjoy-Wolff points.

Abstract

When \ph\ is an analytic self-map of the unit disk with Denjoy-Wolff point $a \in \D$, and $\rho(\W) = \psi(a)$, we give an exact characterization for when \W\ is normaloid. We also determine the spectral radius, essential spectral radius, and essential norm for a class of non-compact composition operators whose symbols have Denjoy-Wolff point in \D. When the Denjoy-Wolff point is on $\partial \D$, we give sufficient conditions for several new classes of normaloid weighted composition operators.