Invertible and normal composition operators on the Hilbert Hardy space of a half-plane

Operators on function spaces of form Cφf = f o φ, where φ is a fixed map are called composition operators with symbol φ. We study such operators acting on the Hilbert Hardy space over the right half-plane and characterize the situations when they are invertible, Fredholm, unitary, and Hermitian. We determine the normal composition operators with inner, respectively with Möbius symbol. In select cases, we calculate their spectra, essential spectra, and numerical ranges.