Cesàro Operators on Rooted Directed Trees
Fuente:
arXiv
Saved in:
| Main Authors: | , , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866915655090962432 |
|---|---|
| author | Abhinand, Mankunikuzhiyil Chavan, Sameer Dharan, Sophiya S. Prasad, Thankarajan |
| author_facet | Abhinand, Mankunikuzhiyil Chavan, Sameer Dharan, Sophiya S. Prasad, Thankarajan |
| contents | In this paper, we introduce and investigate the notion of the Cesáro operator $C_{\mathscr T}$ on a rooted directed tree $\mathscr T.$ When $\mathscr T$ is the rooted tree with no branching vertex, then $C_{\mathscr T}$ is unitarily equivalent to the classical Cesáro operator $C_{0}$ on the sequence space $\ell^2(\mathbb N).$ We prove that for every narrow rooted directed tree $\mathscr T$, $C_{\mathscr T}$ is bounded, with norm bounded above by twice the width of $\mathscr T.$ When the tree is not narrow, this boundedness result no longer holds. Beyond several spectral properties, assuming $\mathscr T$ is leafless and narrow, we show that $C_{\mathscr T}$ is subnormal if and only if $\mathscr T$ is isomorphic to the rooted directed tree without any branching vertex. In particular, this demonstrates that the verbatim analogue of Kriete-Trutt theorem fails in the context of Cesáro operators on rooted directed trees. Nonetheless, under the same hypotheses, $C_{\mathscr T}$ is always a compact perturbation of a subnormal operator. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_00807 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Cesàro Operators on Rooted Directed Trees Abhinand, Mankunikuzhiyil Chavan, Sameer Dharan, Sophiya S. Prasad, Thankarajan Functional Analysis In this paper, we introduce and investigate the notion of the Cesáro operator $C_{\mathscr T}$ on a rooted directed tree $\mathscr T.$ When $\mathscr T$ is the rooted tree with no branching vertex, then $C_{\mathscr T}$ is unitarily equivalent to the classical Cesáro operator $C_{0}$ on the sequence space $\ell^2(\mathbb N).$ We prove that for every narrow rooted directed tree $\mathscr T$, $C_{\mathscr T}$ is bounded, with norm bounded above by twice the width of $\mathscr T.$ When the tree is not narrow, this boundedness result no longer holds. Beyond several spectral properties, assuming $\mathscr T$ is leafless and narrow, we show that $C_{\mathscr T}$ is subnormal if and only if $\mathscr T$ is isomorphic to the rooted directed tree without any branching vertex. In particular, this demonstrates that the verbatim analogue of Kriete-Trutt theorem fails in the context of Cesáro operators on rooted directed trees. Nonetheless, under the same hypotheses, $C_{\mathscr T}$ is always a compact perturbation of a subnormal operator. |
| title | Cesàro Operators on Rooted Directed Trees |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2504.00807 |