$${mathcal {L}}({\mathcal {X}})$$ 中的规范不等式和一个几何常数

IF 1.1 2区 数学 Q1 MATHEMATICS
Pintu Bhunia, Arpita Mal
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引用次数: 0

摘要

我们在({\mathcal {L}}({\mathcal {X}}),\)上引入了一种新的规范(称为(α)规范),({\mathcal {L}}({\mathcal {X}})是定义在有规范线性空间\({\mathcal {X}})上的所有有界线性算子的空间。我们探讨了 \(α \)-规范的各种性质。作为应用,我们得到了算子乘积的数值半径上界,这改进了众所周知的扇形矩阵的数值半径上界。我们利用相应空间单位球的极值点提出了算子的 \(α \)-规范。此外,我们定义了与\({mathcal {X}}\)相关的几何常数(即\(\alpha \)-指数),并研究了\(\alpha \)-指数的性质。特别是,我们得到了一些多面体空间和复希尔伯特空间的(\α \)-指数的精确值。最后,我们研究了规范线性空间的\(ell _p\)-sum的\(alpha \)-index。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Norm inequalities in $${\mathcal {L}}({\mathcal {X}})$$ and a geometric constant

We introduce a new norm (say \(\alpha \)-norm) on \({\mathcal {L}}({\mathcal {X}}),\) the space of all bounded linear operators defined on a normed linear space \({\mathcal {X}}\). We explore various properties of the \(\alpha \)-norm. In addition, we study several equalities and inequalities of the \(\alpha \)-norm of operators on \({\mathcal {X}}.\) As an application, we obtain an upper bound for the numerical radius of product of operators, which improves a well-known upper bound of the numerical radius for sectorial matrices. We present the \(\alpha \)-norm of operators by using the extreme points of the unit ball of the corresponding spaces. Furthermore, we define a geometric constant (say \(\alpha \)-index) associated with \({\mathcal {X}}\) and study properties of the \(\alpha \)-index. In particular, we obtain the exact value of the \(\alpha \)-index for some polyhedral spaces and complex Hilbert space. Finally, we study the \(\alpha \)-index of \(\ell _p\)-sum of normed linear spaces.

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来源期刊
CiteScore
2.00
自引率
8.30%
发文量
67
审稿时长
>12 weeks
期刊介绍: The Banach Journal of Mathematical Analysis (Banach J. Math. Anal.) is published by Birkhäuser on behalf of the Tusi Mathematical Research Group. Banach J. Math. Anal. is a peer-reviewed electronic journal publishing papers of high standards with deep results, new ideas, profound impact, and significant implications in all areas of functional analysis and operator theory and all modern related topics. Banach J. Math. Anal. normally publishes survey articles and original research papers numbering 15 pages or more in the journal’s style. Shorter papers may be submitted to the Annals of Functional Analysis or Advances in Operator Theory.
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