Quasi-hyperbolic semigroups

We introduce the notion of quasi-hyperbolic operators and C0-semigroups. Examples include the push-forward operator associated with a quasi-Anosov diffeomorphism or flow. A quasi-hyperbolic operator can be characterised by a simple spectral property or as the restriction of a hyperbolic operator to an invariant subspace. There is a corresponding spectral property for the generator of a C0-semigroup, and it characterises quasi-hyperbolicity on Hilbert spaces but not on other Banach spaces. We exhibit some weaker properties which are implied by the spectral property.