Linear isometries on the annulus: description and spectral properties

We give a complete characterisation of the linear isometries of $\mathrm{H}\mathrm{o}\mathrm{l}(\mathrm{\Omega })$, where $\mathrm{\Omega }$ 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.