Vasudevarao Allu
580 California St., Suite 400
San Francisco, CA, 94104
A Geometric characterization of Banach spaces with $p$-Bohr radius
2022
https://doi.org/10.48550/ARXIV.2201.03849
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Abstract
For any complex Banach space $X$ and each $p \in [1,\infty)$, we introduce the $p$-Bohr radius of order $N(\in \mathbb{N})$ is $\widetilde{R}_{p,N}(X)$ defined by $$ \widetilde{R}_{p,N}(X)=\sup \left\{r\geq 0: \sum_{k=0}^{N}\norm{x_k}^p r^{pk} \leq \norm{f}^p_{H^{\infty}(\mathbb{D}, X)}\right\}, $$ where $f(z)=\sum_{k=0}^{\infty} x_{k}z^k \in H^{\infty}(\mathbb{D}, X)$. We also introduce the following geometric notion of $p$-uniformly $\mathbb{C}$-convexity of order $N$ for a complex Banach space $X$ for some $N \in \mathbb{N}$. For $p\in [2,\infty)$, a complex Banach space $X$ is called $p$-uniformly $\mathbb{C}$-convex of order $N$ if there exists a constant $\lambda > 0$ such that \begin{align}\label{e-0.1} \left(\norm{x_0}^p + \lambda \norm{x_1}^p + {\lambda}^2 \norm{x_2}^p + \cdots + {\lambda}^N \norm{x_N}^p \right)^{1/p} \leq \max_{\theta \in [0,2\pi)} \norm{x_0 + \sum_{k=1}^{N}e^{i \theta}x_k} \end{align} for all $x_0$, $x_1$,$\dots$, $x_N$ $\in X$. We denote $A_{p,N}(X)$,...
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