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| Format: | Preprint |
| Published: |
2024
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| Online Access: | https://arxiv.org/abs/2410.22006 |
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| _version_ | 1866912112830316544 |
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| author | Bouabdillah, Oualid |
| author_facet | Bouabdillah, Oualid |
| contents | For a subset $E = \{ξ_1, ..., ξ_N\}$ of the unit circle $\mathbb{T}$, the notion of Ritt$_E$ operators on a Banach space and their functional calculus on generalized Stolz domains was developed and studied in arXiv:2203.05373.
In this paper, we define a quadratic functional calculus for a Ritt$_E$ operator on $E_r$, by a decomposition of type Franks-McIntosh. We show that with some hypothesis on the cotype of $X$, this notion is equivalent to the existence of a bounded functional calculus on $E_r$.
We define for a Ritt$_E$ operator on a Banach space $X$ and for any positive real number $α$ and for any $x \in X$
$$
\Vert{x}\Vert_{T,α} = \lim\limits_{n\rightarrow \infty}\Bigl\Vert{\sum\limits_{k=1}^n k^{α- 1/2} \varepsilon_k \otimes T^{k-1}\prod\limits_{j=1}^N(I-\overline{ξ_j}T)^α(x)}\Bigr\Vert_{{\rm Rad}(X)}
$$
We show that, under the condition of finite cotype of $X$, a Ritt$_E$ operator admits a quadratic functional calculus if and only if the estimates $\Vert{x}\Vert_{T,α} \lesssim \Vert{x}\Vert$ hold for both $T$ and $T^*$.
We finally prove the equivalence between these square functions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_22006 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Square functions associated with Ritt$_E$ operators Bouabdillah, Oualid Functional Analysis For a subset $E = \{ξ_1, ..., ξ_N\}$ of the unit circle $\mathbb{T}$, the notion of Ritt$_E$ operators on a Banach space and their functional calculus on generalized Stolz domains was developed and studied in arXiv:2203.05373. In this paper, we define a quadratic functional calculus for a Ritt$_E$ operator on $E_r$, by a decomposition of type Franks-McIntosh. We show that with some hypothesis on the cotype of $X$, this notion is equivalent to the existence of a bounded functional calculus on $E_r$. We define for a Ritt$_E$ operator on a Banach space $X$ and for any positive real number $α$ and for any $x \in X$ $$ \Vert{x}\Vert_{T,α} = \lim\limits_{n\rightarrow \infty}\Bigl\Vert{\sum\limits_{k=1}^n k^{α- 1/2} \varepsilon_k \otimes T^{k-1}\prod\limits_{j=1}^N(I-\overline{ξ_j}T)^α(x)}\Bigr\Vert_{{\rm Rad}(X)} $$ We show that, under the condition of finite cotype of $X$, a Ritt$_E$ operator admits a quadratic functional calculus if and only if the estimates $\Vert{x}\Vert_{T,α} \lesssim \Vert{x}\Vert$ hold for both $T$ and $T^*$. We finally prove the equivalence between these square functions. |
| title | Square functions associated with Ritt$_E$ operators |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2410.22006 |