Local spectral theory for subordinated operators: the Cesàro operator and beyond
Fuente:
arXiv
Salvato in:
| Autori principali: | , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2025
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866908480730824704 |
|---|---|
| author | Gallardo-Gutiérrez, Eva A. González-Doña, F. Javier |
| author_facet | Gallardo-Gutiérrez, Eva A. González-Doña, F. Javier |
| contents | 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 $$\mathcal{H}_ν= \int_{0}^{\infty} T_t\; dν(t)$$ is the subordinated operator associated to $\mathcal{T}$, where $ν$ is a sufficiently regular complex Borel measure supported on $[0,\infty)$, it is shown that $\mathcal{H}_ν$ 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 $\mathcal{H}_ν^{\ast}$ has trivial spectral subspaces and enjoys the Dunford property. As an application, for the classical Cesàro operator $\mathcal{C}$ acting on the Hardy spaces $H^p$ ($1<p<\infty$), 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_04376 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Local spectral theory for subordinated operators: the Cesàro operator and beyond Gallardo-Gutiérrez, Eva A. González-Doña, F. Javier Functional Analysis 47A11, 47B33, 47B38 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 $$\mathcal{H}_ν= \int_{0}^{\infty} T_t\; dν(t)$$ is the subordinated operator associated to $\mathcal{T}$, where $ν$ is a sufficiently regular complex Borel measure supported on $[0,\infty)$, it is shown that $\mathcal{H}_ν$ 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 $\mathcal{H}_ν^{\ast}$ has trivial spectral subspaces and enjoys the Dunford property. As an application, for the classical Cesàro operator $\mathcal{C}$ acting on the Hardy spaces $H^p$ ($1<p<\infty$), 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. |
| title | Local spectral theory for subordinated operators: the Cesàro operator and beyond |
| topic | Functional Analysis 47A11, 47B33, 47B38 |
| url | https://arxiv.org/abs/2508.04376 |