Some class of numerical radius peak $n$-linear mappings on $l_p$-spaces

Q3 Mathematics Matematychni Studii Pub Date : 2022-03-31 DOI:10.30970/ms.57.1.10-15
S. Kim
{"title":"Some class of numerical radius peak $n$-linear mappings on $l_p$-spaces","authors":"S. Kim","doi":"10.30970/ms.57.1.10-15","DOIUrl":null,"url":null,"abstract":"For $n\\geq 2$ and a real Banach space $E,$ ${\\mathcal L}(^n E:E)$ denotes the space of all continuous $n$-linear mappings from $E$ to itself.Let $$\\Pi(E)=\\Big\\{[x^*, (x_1, \\ldots, x_n)]: x^{*}(x_j)=\\|x^{*}\\|=\\|x_j\\|=1~\\mbox{for}~{j=1, \\ldots, n}\\Big\\}.$$For $T\\in {\\mathcal L}(^n E:E),$ we define $$\\qopname\\relax o{Nr}({T})=\\Big\\{[x^*, (x_1, \\ldots, x_n)]\\in \\Pi(E): |x^{*}(T(x_1, \\ldots, x_n))|=v(T)\\Big\\},$$where $v(T)$ denotes the numerical radius of $T$.$T$ is called {\\em numerical radius peak mapping} if there is $[x^{*}, (x_1, \\ldots, x_n)]\\in \\Pi(E)$ such that $\\qopname\\relax o{Nr}({T})=\\{\\pm [x^{*}, (x_1, \\ldots, x_n)]\\}.$In this paper, we investigate some class of numerical radius peak mappings in ${\\mathcalL}(^n l_p:l_p)$ for $1\\leq p<\\infty.$ Let $(a_{j})_{j\\in \\mathbb{N}}$ be a bounded sequence in $\\mathbb{R}$ such that $\\sup_{j\\in \\mathbb{N}}|a_j|>0.$Define $T\\in {\\mathcal L}(^n l_p:l_p)$ by$$T\\Big(\\sum_{i\\in \\mathbb{N}}x_i^{(1)}e_i, \\cdots, \\sum_{i\\in \\mathbb{N}}x_i^{(n)}e_i \\Big)=\\sum_{j\\in \\mathbb{N}}a_{j}~x_{j}^{(1)}\\cdots x_{j}^{(n)}~e_j.\\qquad\\eqno(*)$$In particular is proved the following statements:\\$1.$\\ If $1< p<+\\infty$ then $T$ is a numerical radius peak mapping if and only if there is $j_0\\in \\mathbb{N}$ such that$$|a_{j_0}|>|a_{j}|~\\mbox{for every}~j\\in \\mathbb{N}\\backslash\\{j_0\\}.$$ \n$2.$\\ If $p=1$ then $T$ is not a numerical radius peak mapping in ${\\mathcal L}(^n l_1:l_1).$","PeriodicalId":37555,"journal":{"name":"Matematychni Studii","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2022-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Matematychni Studii","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.30970/ms.57.1.10-15","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"Mathematics","Score":null,"Total":0}
引用次数: 0

Abstract

For $n\geq 2$ and a real Banach space $E,$ ${\mathcal L}(^n E:E)$ denotes the space of all continuous $n$-linear mappings from $E$ to itself.Let $$\Pi(E)=\Big\{[x^*, (x_1, \ldots, x_n)]: x^{*}(x_j)=\|x^{*}\|=\|x_j\|=1~\mbox{for}~{j=1, \ldots, n}\Big\}.$$For $T\in {\mathcal L}(^n E:E),$ we define $$\qopname\relax o{Nr}({T})=\Big\{[x^*, (x_1, \ldots, x_n)]\in \Pi(E): |x^{*}(T(x_1, \ldots, x_n))|=v(T)\Big\},$$where $v(T)$ denotes the numerical radius of $T$.$T$ is called {\em numerical radius peak mapping} if there is $[x^{*}, (x_1, \ldots, x_n)]\in \Pi(E)$ such that $\qopname\relax o{Nr}({T})=\{\pm [x^{*}, (x_1, \ldots, x_n)]\}.$In this paper, we investigate some class of numerical radius peak mappings in ${\mathcalL}(^n l_p:l_p)$ for $1\leq p<\infty.$ Let $(a_{j})_{j\in \mathbb{N}}$ be a bounded sequence in $\mathbb{R}$ such that $\sup_{j\in \mathbb{N}}|a_j|>0.$Define $T\in {\mathcal L}(^n l_p:l_p)$ by$$T\Big(\sum_{i\in \mathbb{N}}x_i^{(1)}e_i, \cdots, \sum_{i\in \mathbb{N}}x_i^{(n)}e_i \Big)=\sum_{j\in \mathbb{N}}a_{j}~x_{j}^{(1)}\cdots x_{j}^{(n)}~e_j.\qquad\eqno(*)$$In particular is proved the following statements:\$1.$\ If $1< p<+\infty$ then $T$ is a numerical radius peak mapping if and only if there is $j_0\in \mathbb{N}$ such that$$|a_{j_0}|>|a_{j}|~\mbox{for every}~j\in \mathbb{N}\backslash\{j_0\}.$$ $2.$\ If $p=1$ then $T$ is not a numerical radius peak mapping in ${\mathcal L}(^n l_1:l_1).$
查看原文
分享 分享
微信好友 朋友圈 QQ好友 复制链接
本刊更多论文
l_p$-空间上的一类数值半径峰$n$-线性映射
对于$n\geq 2$和实巴拿赫空间$E,$ ${\数学L}(^n E:E)$表示从$E$到自身的所有连续$n$线性映射的空间。让$ $ \大π(E) = \ \ {[x ^ * (x_1、\ ldots x_n)]: x ^ {*} (x_j) = \ | x ^ {*} \ | = \ | x_j \ | = 1 ~ \ mbox {} ~ {\ ldots j = 1, n} \大\}。$ $ $ T \ {\ mathcal L} (^ n E: E),我们定义美元美元\ qopname \放松o {Nr} ({T}) =大\ \ {[x ^ * (x_1、\ ldots x_n)] \ \π(E): | x ^ {*} (T (x_1、\ ldots x_n)) | = v (T)大\ \},在v (T)美元美元表示的数值半径$ T $。T美元被称为{\ em数值半径峰映射}如果有美元[x ^ {*}, (x_1、\ ldots x_n)] \ \π(E),这样美元\ qopname \放松o {Nr} ({T}) = \{下午\ [x ^ {*}, (x_1、\ ldots x_n)] \}。本文研究了${\mathcalL}(^n l_p:l_p)$中$1\leq p0的一类数值半径峰映射。定义T美元在{\ mathcal L} \ (^ n l_p: l_p) $ $ $ T \大(\ sum_{我\ \ mathbb {n}} x_i ^ {(1)} e_i, \ cdots \ sum_{我\ \ mathbb {n}} x_i ^ {(n)} e_i \大)= \ sum_ {j \ \ mathbb {n}}现代{j} ~间{j} ^ {(1)} \ cdots间{j} ^ {(n)} ~ e_j。\ qquad \ eqno(*) $ $特别是证明下列语句:\ 1美元。\如果美元1 < p |现代{j} | ~ \ mbox{每}~ j \ \ mathbb {N} \反斜杠\ {j_0 \}。$ $ $ 2。$\如果$p=1$,则$T$不是${\mathcal L}(^n l_1:l_1)中的数值半径峰映射
本文章由计算机程序翻译,如有差异,请以英文原文为准。
求助全文
约1分钟内获得全文 去求助
来源期刊
Matematychni Studii
Matematychni Studii Mathematics-Mathematics (all)
CiteScore
1.00
自引率
0.00%
发文量
38
期刊介绍: Journal is devoted to research in all fields of mathematics.
期刊最新文献
On the h-measure of an exceptional set in Fenton-type theorem for Taylor-Dirichlet series Almost periodic distributions and crystalline measures Reflectionless Schrodinger operators and Marchenko parametrization Existence of basic solutions of first order linear homogeneous set-valued differential equations Real univariate polynomials with given signs of coefficients and simple real roots
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
现在去查看 取消
×
提示
确定
0
微信
客服QQ
Book学术公众号 扫码关注我们
反馈
×
意见反馈
请填写您的意见或建议
请填写您的手机或邮箱
已复制链接
已复制链接
快去分享给好友吧!
我知道了
×
扫码分享
扫码分享
Book学术官方微信
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术
文献互助 智能选刊 最新文献 互助须知 联系我们:info@booksci.cn
Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。
Copyright © 2023 Book学术 All rights reserved.
ghs 京公网安备 11010802042870号 京ICP备2023020795号-1