Extensions of the operator Bellman and operator Holder type inequalities

IF 16.4 1区 化学 Q1 CHEMISTRY, MULTIDISCIPLINARY
M. Bakherad, F. Kittaneh
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引用次数: 0

Abstract

In this paper, we employ the concept of operator means as well as some operator techniques to establish new operator Bellman and operator H\"{o}lder type inequalities. Among other results, it is shown that if $\mathbf{A}=(A_t)_{t\in \Omega}$ and $\mathbf{B}=(B_t)_{t\in \Omega}$ are continuous fields of positive invertible operators in a unital $C^*$-algebra ${\mathscr A}$ such that $\int_{\Omega}A_t\,d\mu(t)\leq I_{\mathscr A}$ and $\int_{\Omega}B_t\,d\mu(t)\leq I_{\mathscr A}$, and if $\omega_f$ is an arbitrary operator mean with the representing function $f$, then \begin{align*} \left(I_{\mathscr A}-\int_{\Omega}(A_t \omega_f B_t)\,d\mu(t)\right)^p \geq\left(I_{\mathscr A}-\int_{\Omega}A_t\,d\mu(t)\right) \omega_{f^p}\left(I_{\mathscr A}-\int_{\Omega}B_t\,d\mu(t)\right) \end{align*} for all $0 < p \leq 1$, which is an extension of the operator Bellman inequality.
算子贝尔曼不等式和算子霍尔德不等式的扩展
在本文中,我们运用算子手段的概念以及一些算子技术,建立了新的算子贝尔曼不等式和算子霍德尔不等式。除其他结果外,本文还证明了如果 $\mathbf{A}=(A_t)_{t\in \Omega}$ 和 $\mathbf{B}=(B_t)_{t\in \Omega}$ 是一元 $C^*$ 代数 ${\mathscr A}$ 中正可逆算子的连续域,使得 $\int_{\Omega}A_t\、和 $int_{{Omega}B_t\,d\mu(t)\leq I_{{mathscr A}$,并且如果 $omega_f$ 是一个任意算子均值与代表函数 $f$,那么 \begin{align*}\left(I_{\mathscr A}-\int_{\Omega}(A_t \omega_f B_t)\,d\mu(t)\right)^p \geq\left(I_{\mathscr A}-\int_{\Omega}A_t\、d\mu(t)\right) \omega_{f^p}\left(I_{\mathscr A}-\int_{\Omega}B_t\,d\mu(t)\right) \end{align*} for all $0 < p \leq 1$,这是算子贝尔曼不等式的扩展。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
Accounts of Chemical Research
Accounts of Chemical Research 化学-化学综合
CiteScore
31.40
自引率
1.10%
发文量
312
审稿时长
2 months
期刊介绍: Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance. Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.
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