Local spectral theory for subordinated operators: the Ces\`aro operator and beyond

TL;DR

This paper investigates local spectral properties of subordinated operators derived from $C_0$-semigroups, including the Cesàro operator, revealing their spectral behavior and properties like SVEP and Dunford property, with applications to Hardy spaces.

Contribution

It provides new insights into the local spectral theory of subordinated operators, especially the Cesàro operator, and characterizes their spectral properties based on the underlying semigroup geometry.

Findings

Subordinated operators lack SVEP and have dense local spectral subspaces.

The adjoint operators have trivial spectral subspaces and satisfy Dunford property.

The local spectrum of the Cesàro operator on Hardy spaces coincides with its spectrum.

Abstract

We study local spectral properties for subordinated operators arising from $C_0$-semigroups. Specifically, if $\mathcal{T}=(T_t)_{t\geq 0}$ is a $C_0$-semigroup acting boundedly on a complex Banach space and $$\mathcal{H}_\nu = \int_{0}^{\infty} T_t\; d\nu(t)$$ is the subordinated operator associated to $\mathcal{T}$, where $\nu$ is a sufficiently regular complex Borel measure supported on $[0,\infty)$, 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 Ces\`aro operator $\mathcal{C}$ acting on the Hardy spaces $H^p$ ($1<p<\infty$), 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 $H^p$ ($1<p<\infty$), which will depend only on the geometry of the associated Koenigs domain.