{"title":"Extensions of the operator Bellman and operator Holder type inequalities","authors":"M. Bakherad, F. Kittaneh","doi":"10.33205/cma.1435944","DOIUrl":null,"url":null,"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\n \\begin{align*}\n \\left(I_{\\mathscr A}-\\int_{\\Omega}(A_t \\omega_f B_t)\\,d\\mu(t)\\right)^p\n \\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)\n \\end{align*}\n for all $0 < p \\leq 1$, which is an extension of the operator Bellman inequality.","PeriodicalId":1,"journal":{"name":"Accounts of Chemical Research","volume":"11 1","pages":""},"PeriodicalIF":16.4000,"publicationDate":"2024-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Accounts of Chemical Research","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.33205/cma.1435944","RegionNum":1,"RegionCategory":"化学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"CHEMISTRY, MULTIDISCIPLINARY","Score":null,"Total":0}
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.
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.