Bohr 和 Rogosinski 不等式对于算子值全纯函数
Bohr and Rogosinski inequalities for operator valued holomorphic functions
AI总结：
本文研究了复Banach空间中算子值全纯函数的Bohr半径和p-均匀C-凸性，证明了p-均匀C-凸性等价于Bohr半径正数，并探讨了Lebesgue空间中的相关性质。
摘要：
对于任何复Banach空间X和每个p∈[1,∞)，我们引入了p-Bohr半径的阶数N（N∈N）定义为$$ \widetilde{R}_{p,N}(X)=\sup \left\{r\geq 0: \sum_{k=0}^{N}\norm{x_k}^p r^{pk} \leq \norm{f}^p_{H^{\infty}(\mathbb{D}, X)}\right\}, $$其中f(z)=∑_{k=0}^∞x_kz^k∈H^∞(D,X)。这里D={z∈C:|z|<1}表示单位圆盘。我们还引入了复Banach空间X的p-均匀C-凸性阶数N的几何概念。在本文中，对于p∈[2,∞)和每个N∈N，我们证明了复Banach空间X是p-均匀C-凸性阶数N当且仅当p-Bohr半径阶数N$$ \widetilde{R}_{p,N}(X)>0。我们还研究了Lebesgue空间L^q(μ)中p-Bohr半径阶数N的性质。最后，我们证明了有界全纯函数从单位圆盘D到B(H)的算子值版本的Bohr和Rogosinski不等式精炼版本。
英文摘要：
For any complex Banach space $X$ and each $p \in [1,\infty)$, we introduce the $p$-Bohr radius of order $N(\in \mathbb{N})$ is $\widetilde{R}_{p,N}(X)$ defined by $$ \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 $f(z)=\sum_{k=0}^{\infty} x_{k}z^k \in H^{\infty}(\mathbb{D}, X)$. Here $\mathbb{D}= \{z\in \mathbb{C}: |z| <1\}$ denotes the unit disk. We also introduce the following geometric notion of $p$-uniformly $\mathbb{C}$-convexity of order $N$ for a complex Banach space $X$ for some $N \in \mathbb{N}$. In this paper, for $p\in [2,\infty)$ and each $N \in \mathbb{N}$, we prove that a complex Banach space $X$ is $p$-uniformly $\mathbb{C}$-convex of order $N$ if, and only if, the $p$-Bohr radius of order $N$ $\widetilde{R}_{p,N}(X)>0$. We also study the $p$-Bohr radius of order $N$ for the Lebesgue spaces $L^q (μ)$ for $1\leq p<q<\infty$ or $1\leq q \leq p <2$. Finally, we prove an operator valued analogue of a refined version of Bohr and Rogosinski inequality for bounded holomorphic functions from the unit disk $\mathbb{D}$ into $\mathcal{B(\mathcal{H})}$, where $\mathcal{B(\mathcal{H})}$ denotes the space of all bounded linear operator on a complex Hilbert space $\mathcal{H}$.
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