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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2410.20435 |
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Table of Contents:
- Let $α>0$ and $μ$ be a positive Borel measure on the interval $[0,1)$. The Hankel matrix $\mathcal{H}_{μ,α}=(μ_{n,k,α})_{n,k\ge0}$ with entries $μ_{n,k,α}=\int_{[0,1)}^{}\frac{Γ(n+α)}{Γ(n+1)Γ(α)}t^{n+k}dμ(t)$, induces, formally, the generalized-Hilbert operator as $$ \mathcal{H}_{μ,α}\left ( f \right ) \left ( z \right ) =\sum_{n=0}^{\infty} \left (\sum_{k=0}^{\infty} μ_{n,k,α}a_k \right )z^n,z\in\mathbb{D} $$ where $f(z)={\textstyle \sum_{k=0}^{\infty }} a_kz^k$ is an analytic function in $\mathbb{D}$. This article is devoted study the measures $μ$ for which $\mathcal{H}_{μ,α}$ is a bounded(resp., compact) operator from $H^p(0<p\le1)$ into $H^p(1\le q<\infty)$. Then, we also study the analogous problem in the Hardy spaces $H^p(1\le p\le2)$. Finally, we obtain the essential norm of $\mathcal{H}_{μ,α}$ from $H^p(0<p\le1)$ into $H^p(1\le q<\infty)$.