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

We study local spectral properties for subordinated operators arising from $C_0$-semigroups. Specifically, if $\mathcal{T}=(T_t{)}_{t\ge 0}$ is a $C_0$-semigroup acting boundedly on a complex Banach space and $${\mathcal{H}}_{\nu }={\int }_0^{\mathrm{\infty }}{T}_{t}d\nu (t)$$ is the subordinated operator associated to $\mathcal{T}$, where $\nu$ is a sufficiently regular complex Borel measure supported on $[0,\mathrm{\infty })$, it is shown that ${\mathcal{H}}_{\nu }$ does not enjoy the Single Valued Extension Property (SVEP) and has dense glocal spectral subspaces in terms of the spectrum of the generator of $\mathcal{T}$. Likewise, the adjoint ${\mathcal{H}}_{\nu }^{\ast }$ has trivial spectral subspaces and enjoys the Dunford property. As an application, for the classical Cesàro operator $\mathcal{C}$ acting on the Hardy spaces $H^p$ ($1<p<\mathrm{\infty }$), it follows that the local spectrum of $\mathcal{C}$ at any non-zero $H^p$-function or the spectrum of the restriction of $\mathcal{C}$ to any of its non-trivial closed invariant subspaces coincides with the spectrum of $\mathcal{C}$. Finally, we characterize the local spectral properties of subordinated operators arising from hyperbolic semigroups of composition operators acting on $H^p$ ($1<p<\mathrm{\infty }$), which will depend only on the geometry of the associated Koenigs domain.