Local spectral theory for subordinated operators: The Cesàro operator and beyond

We study local spectral properties for subordinated operators arising from ‐semigroups. Specifically, if is a ‐semigroup acting boundedly on a complex Banach space and is the subordinated operator associated to , where is a sufficiently regular complex Borel measure supported on , it is shown that does not enjoy the single valued extension property (SVEP) and has dense glocal spectral subspaces in terms of the spectrum of the generator of . Likewise, the adjoint has trivial spectral subspaces and enjoys the Dunford property. As an application, for the classical Cesàro operator acting on the Hardy spaces (), it follows that the local spectrum of at any non‐zero ‐function or the spectrum of the restriction of to any of its non‐trivial closed invariant subspaces coincides with the spectrum of . Finally, we characterize the local spectral properties of subordinated operators arising from hyperbolic semigroups of composition operators acting on (), which will depend only on the geometry of the associated Koenigs domain.