Linear isometries on the annulus: description and spectral properties

TL;DR

This paper characterizes all linear isometries on holomorphic function spaces over certain domains, introduces new methods for holomorphic maps preserving the unit circle, and analyzes their spectral properties.

Contribution

It provides a complete description of linear isometries on holomorphic functions over the half-plane, plane, and annulus, with novel techniques for maps preserving the unit circle and spectral analysis.

Findings

Complete characterization of linear isometries on specified domains

New techniques for holomorphic maps preserving the unit circle

Spectral properties of isometries on the annulus

Abstract

We give a complete characterisation of the linear isometries of ${\rm Hol}(\Omega)$, where $\Omega$ is the half-plane, the complex plane or an annulus centered at 0 and symmetric to the unit circle. Moreover, we introduce new techniques to describe the holomorphic maps on the annulus that preserve the unit circle, and we finish by proving results about the spectra of the linear isometries on the annulus.