A note on composition operators in a half-plane

TL;DR

This paper extends known results about composition operators from the disk to the upper half-plane, providing conditions for closed range and isometry similarity, and showing such operators can be similar to isometries even if not inner functions.

Contribution

It establishes new criteria for composition operators on the Hardy space of the upper half-plane and demonstrates their similarity to isometries beyond inner functions.

Findings

Conditions for closed range of composition operators in the half-plane

Criteria for composition operators to be similar to isometries

Examples of composition operators similar to isometries despite non-inner symbols

Abstract

Conditions for a composition operator on the Hardy space of the disk to have closed range or be similar to an isometry are well known. We provide such conditions for composition operators on the Hardy space of the upper half-plane. We also show that the operator of composition with an analytic self-map \Phi\ of the upper half-plane can be similar to an isometry even when \Phi\ is far from being an inner function.