论文标题
Bohr和Rogosinski的操作员有价值的全态功能
Bohr and Rogosinski inequalities for operator valued holomorphic functions
论文作者
论文摘要
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 $$ \widetilde{R}_{p,N}(X)=\sup \ left \ {r \ geq 0：\ sum_ {k = 0}^{n} \ norm {x_k}^p r^{pk} \ leq \ norm {f}^p_ { $ f（z）= \ sum_ {k = 0}^{\ infty} x_ {k} z^k \ in H^{\ infty}（\ Mathbb {d}，x）$。这里$ \ mathbb {d} = \ {z \ in \ mathbb {c}：| z | | <1 \} $表示单位磁盘。我们还介绍了以下几何概念$ p $ - 均匀的$ \ mathbb {c} $ - 订单$ n $的复杂Banach Space $ x $的订单$ n $，用于某些$ n \ in \ Mathbb {n} $。在本文中，对于$ p \ in [2，\ infty）$而每个$ n \ in \ mathbb {n} $，我们证明了一个复杂的banach space $ x $是$ p $ p $ p $ - 均$ \ mathbb {c} $ convex $ n $，并且只有$ n $，并且只有$ p $ n $ of p $ n $ of-p $ n $ $ \ widetilde {r} _ {p，n}（x）> 0 $。我们还研究了Lebesgue Spaces $ l^Q（μ）$ $ p $ -bohr半径$ n $，价格为$ 1 \ leq p <q <q <\ infty $或$ 1 \ leq q \ leq q \ leq p <2 $。最后，我们证明了一个经营者的bohr和rogosinski不平等的经营者，从单位磁盘$ \ mathbb {d} $中的有限型全体形态函数中的不平等现象中有价值的类似物复杂的Hilbert Space $ \ Mathcal {H} $上的线性操作员。
For any complex Banach space $X$ and each $p \in [1,\infty)$, we introduce the $p$-Bohr radius of order $N(\in \mathbb{N})$ is $\widetilde{R}_{p,N}(X)$ defined by $$ \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 $\mathcal{H}$.
