巴拿赫空间的扩张理论方法

Swapan Jana, Sourav Pal, Saikat Roy
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引用次数: 0

摘要

对于复巴纳赫空间 $\mathbb X$,我们证明,当且仅当 $\mathbb X$ 上的每一个严格收缩 $T$ 都能稀释为等势时,$\mathbb X$ 是一个希尔伯特空间,当且仅当 $\mathbb X$ 上的每一个严格收缩 $T$ 都能稀释为等势时,函数 $A_T:\由$A_T(x)=(\|x\|^2-\|Tx\|^2)^\{frac{1}{2}}$定义的函数$A_T: \rightarrow [0, \infty]$在$\mathbb X$上给出了一个规范。我们还发现了其他几个必要条件和充分条件,从而使巴拿赫空间成为希尔伯特空间。我们构建了非希尔伯特-巴拿赫空间上不扩张为等距的严格收缩的例子。然后,我们描述了非希尔伯特-巴拿赫空间上所有扩张为等距的严格收缩的特征,并为它们找到了明确的等距扩张。我们还证明了其他一些结果,包括巴拿赫空间互补子空间的特征、沃尔德等距扩展到巴拿赫空间单元以及用扩张描述巴拿赫空间算子的规范达到集。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
A dilation theoretic approach to Banach spaces
For a complex Banach space $\mathbb X$, we prove that $\mathbb X$ is a Hilbert space if and only if every strict contraction $T$ on $\mathbb X$ dilates to an isometry if and only if for every strict contraction $T$ on $\mathbb X$ the function $A_T: \mathbb X \rightarrow [0, \infty]$ defined by $A_T(x)=(\|x\|^2 -\|Tx\|^2)^{\frac{1}{2}}$ gives a norm on $\mathbb X$. We also find several other necessary and sufficient conditions in this thread such that a Banach sapce becomes a Hilbert space. We construct examples of strict contractions on non-Hilbert Banach spaces that do not dilate to isometries. Then we characterize all strict contractions on a non-Hilbert Banach space that dilate to isometries and find explicit isometric dilation for them. We prove several other results including characterizations of complemented subspaces in a Banach space, extension of a Wold isometry to a Banach space unitary and describing norm attainment sets of Banach space operators in terms of dilations.
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