Normal weighted composition operators on the Hardy space

TL;DR

This paper characterizes unitary and normal weighted composition operators on the Hardy space H^2, revealing conditions on the inducing functions g and h, and providing spectral descriptions for these operators.

Contribution

It provides a complete characterization of unitary and normal weighted composition operators on H^2, including conditions on g and h, and spectra analysis.

Findings

Every automorphism of U induces a unitary weighted composition operator.

Normality of W_{h,g} requires g to be univalent or constant.

Spectral properties are described for unitary and normal cases.

Abstract

Let g be an analytic function on the open unit disc U such that g(U) is contained in U, and let h be an analytic function on U such that the weighted composition operator W_{h,g) defined by W_{h,g}f = h f(g) is bounded on the Hardy space H^2. We characterize those weighted composition operators on H^2 that are unitary, showing that in contrast to the unweighted case (h=1), every automorphism of U induces a unitary weighted composition operator. A conjugation argument, using these unitary operators, allows us to describe all normal weighted composition operators on H^2 for which the inducing map g fixes a point in U. This description shows both h and g must be linear fractional in order for W_{h,g} to be normal (assuming g fixes a point in U). In general, we show that if W_{h, g} is normal on H^2 and h is not the zero function, then g must be either univalent on U or constant. Descriptions of spectra are provided for the operator W_{h,g} when it is unitary or when it is normal and g fixes a point in U.