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

Abstract

We study local spectral properties for subordinated operators arising from C0-semigroups. Specifically, if T = (Tt)t≥0 is a C0-semigroup acting boundedly on a complex Banach space and Hν is the subordinated operator associated to T , where ν is a sufficiently regular complex Borel measure supported on [0,∞), it is shown that Hν does not enjoy the Single Valued Extension Property (SVEP) and has dense glocal spectral subspaces in terms of the spectrum of the generator of T . Likewise, the adjoint H∗ν has trivial spectral subspaces and enjoys the Dunford property. As an application, for the classical Cesàro operator C acting on the Hardy spaces Hp (1 < p < ∞), it follows that the local spectrum of C at any non-zero Hp-function or the spectrum of the restriction of C to any of its nontrivial closed invariant subspaces coincides with the spectrum of C. Finally, we characterize the local spectral properties of subordinated operators arising from hyperbolic semigroups of composition operators acting on Hp (1 < p < ∞), which will depend only on the geometry of the associated Koenigs domain.