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Banach Journal of Mathematical Analysis最新文献

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The generalized polar decomposition, the weak complementarity and the parallel sum for adjointable operators on Hilbert $$C^*$$ -modules 希尔伯特 $$C^*$$ 模块上可相邻算子的广义极性分解、弱互补性和平行和
IF 1.2 2区 数学
Banach Journal of Mathematical Analysis Pub Date : 2024-05-11 DOI: 10.1007/s43037-024-00351-z
Xiaofeng Zhang, Xiaoyi Tian, Qingxiang Xu
{"title":"The generalized polar decomposition, the weak complementarity and the parallel sum for adjointable operators on Hilbert $$C^*$$ -modules","authors":"Xiaofeng Zhang, Xiaoyi Tian, Qingxiang Xu","doi":"10.1007/s43037-024-00351-z","DOIUrl":"https://doi.org/10.1007/s43037-024-00351-z","url":null,"abstract":"<p>This paper deals mainly with some aspects of the adjointable operators on Hilbert <span>(C^*)</span>-modules. A new tool called the generalized polar decomposition for each adjointable operator is introduced and clarified. As an application, the general theory of the weakly complementable operators is set up in the framework of Hilbert <span>(C^*)</span>-modules. It is proved that there exists an operator equation which has a unique solution, whereas this unique solution fails to be the reduced solution. Some investigations are also carried out in the Hilbert space case. It is proved that there exist a closed subspace <i>M</i> of certain Hilbert space <i>K</i> and an operator <span>(Tin {mathbb {B}}(K))</span> such that <i>T</i> is (<i>M</i>, <i>M</i>)-weakly complementable, whereas <i>T</i> fails to be (<i>M</i>, <i>M</i>)-complementable. The solvability of the equation </p><span>$$begin{aligned} A:B=X^*AX+(I-X)^*B(I-X) quad big (Xin {mathbb {B}}(H)big ) end{aligned}$$</span><p>is also dealt with in the Hilbert space case, where <span>(A,Bin {mathbb {B}}(H))</span> are two general positive operators, and <i>A</i> : <i>B</i> denotes their parallel sum. Among other things, it is shown that there exist certain positive operators <i>A</i> and <i>B</i> on the Hilbert space <span>(ell ^2({mathbb {N}})oplus ell ^2({mathbb {N}}))</span> such that the above equation has no solution.</p>","PeriodicalId":55400,"journal":{"name":"Banach Journal of Mathematical Analysis","volume":"27 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2024-05-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140941051","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Common spectral properties and $$nu $$ -convergence 共谱特性和 $$nu $$ 收敛性
IF 1.2 2区 数学
Banach Journal of Mathematical Analysis Pub Date : 2024-05-09 DOI: 10.1007/s43037-024-00350-0
Soufiane Hadji, Hassane Zguitti
{"title":"Common spectral properties and $$nu $$ -convergence","authors":"Soufiane Hadji, Hassane Zguitti","doi":"10.1007/s43037-024-00350-0","DOIUrl":"https://doi.org/10.1007/s43037-024-00350-0","url":null,"abstract":"<p>In this paper we show that if <span>({T_n})</span> is a sequence of bounded linear operators on a complex Banach space <i>X</i> which <span>(nu )</span>-converges to two different bounded linear operators <i>T</i> and <i>U</i>, then <i>T</i> and <i>U</i> have the same parts of the spectrum. In particular, we generalize the results of Sánchez-Perales and Djordjević (J Math Anal Appl 433:405–415, 2016) and of Ammar (Indag Math 28:424–435, 2017). We also investigate the spectral <span>(nu )</span>-continuity for the surjective spectrum.</p>","PeriodicalId":55400,"journal":{"name":"Banach Journal of Mathematical Analysis","volume":"14 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2024-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140941178","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Norm of the sum of two orthogonal projections 两个正交投影之和的常模
IF 1.2 2区 数学
Banach Journal of Mathematical Analysis Pub Date : 2024-05-06 DOI: 10.1007/s43037-024-00347-9
Cristian Conde
{"title":"Norm of the sum of two orthogonal projections","authors":"Cristian Conde","doi":"10.1007/s43037-024-00347-9","DOIUrl":"https://doi.org/10.1007/s43037-024-00347-9","url":null,"abstract":"<p>In this note, we give a new proof of the following well-known norm formula which holds for any two orthogonal projections <span>(P_{mathcal {T}}, P_{mathcal {S}})</span> on a Hilbert <span>({mathcal {H}},)</span></p><span>$$begin{aligned} Vert P_{mathcal {T}}+P_{mathcal {S}}Vert = 1+Vert P_{mathcal {T}}P_{mathcal {S}}Vert , end{aligned}$$</span><p>unless <span>(P_{mathcal {T}}=P_{mathcal {S}}=0.)</span> This equality was proved by Duncan and Taylor (Proc R Soc Edinb Sect A 75(2):119–129, 1975). We derive this formula from the relationship between the spectra of the sum and product of any two idempotents, as well as various norm inequalities for positive operators defined on <span>({mathcal {H}}.)</span> Applications of our results are given.</p>","PeriodicalId":55400,"journal":{"name":"Banach Journal of Mathematical Analysis","volume":"29 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2024-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140885722","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Multipliers and equivalence of functions, spaces, and operators 函数、空间和算子的乘法和等价性
IF 1.2 2区 数学
Banach Journal of Mathematical Analysis Pub Date : 2024-05-05 DOI: 10.1007/s43037-024-00349-7
M. C. Câmara, C. Carteiro, W. T. Ross
{"title":"Multipliers and equivalence of functions, spaces, and operators","authors":"M. C. Câmara, C. Carteiro, W. T. Ross","doi":"10.1007/s43037-024-00349-7","DOIUrl":"https://doi.org/10.1007/s43037-024-00349-7","url":null,"abstract":"<p>By interpreting the well-known Brown–Halmos theorem for Toeplitz operators in terms of multipliers, we formulate a Brown–Halmos analogue for the product of generalized Toeplitz operators, defined as compressions of multiplication operators to closed subspaces of <span>(L^2({mathbb {T}}))</span>. We use this to define equivalences between two operators in that class by means of multipliers between the spaces where they act. Necessary and sufficient conditions for such an equivalence to be unitary or a similarity are established. The results are applied to Toeplitz and Hankel operators, truncated Toeplitz operators, and dual truncated Toeplitz operators.</p>","PeriodicalId":55400,"journal":{"name":"Banach Journal of Mathematical Analysis","volume":"56 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2024-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140885862","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Marcinkiewicz–Zygmund inequalities in variable Lebesgue spaces 可变 Lebesgue 空间中的 Marcinkiewicz-Zygmund 不等式
IF 1.2 2区 数学
Banach Journal of Mathematical Analysis Pub Date : 2024-05-04 DOI: 10.1007/s43037-024-00344-y
Marcos Bonich, Daniel Carando, Martin Mazzitelli
{"title":"Marcinkiewicz–Zygmund inequalities in variable Lebesgue spaces","authors":"Marcos Bonich, Daniel Carando, Martin Mazzitelli","doi":"10.1007/s43037-024-00344-y","DOIUrl":"https://doi.org/10.1007/s43037-024-00344-y","url":null,"abstract":"<p>We study <span>(ell ^r)</span>-valued extensions of linear operators defined on Lebesgue spaces with variable exponent. Under some natural (and usual) conditions on the exponents, we characterize <span>(1le rle infty )</span> such that every bounded linear operator <span>(T:L^{q(cdot )}(Omega _2, mu )rightarrow L^{p(cdot )}(Omega _1, nu ))</span> has a bounded <span>(ell ^r)</span>-valued extension. We consider both non-atomic measures and measures with atoms and show the differences that can arise. We present some applications of our results to weighted norm inequalities of linear operators and vector-valued extensions of fractional operators with rough kernel.</p>","PeriodicalId":55400,"journal":{"name":"Banach Journal of Mathematical Analysis","volume":"115 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2024-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140885718","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Interpolation and non-dilatable families of $$mathcal {C}_{0}$$ -semigroups $$mathcal {C}_{0}$ -semigroups 的插值和不可稀疏族
IF 1.2 2区 数学
Banach Journal of Mathematical Analysis Pub Date : 2024-04-18 DOI: 10.1007/s43037-023-00320-y
Raj Dahya
{"title":"Interpolation and non-dilatable families of $$mathcal {C}_{0}$$ -semigroups","authors":"Raj Dahya","doi":"10.1007/s43037-023-00320-y","DOIUrl":"https://doi.org/10.1007/s43037-023-00320-y","url":null,"abstract":"<p>We generalise a technique of Bhat and Skeide (J Funct Anal 269:1539–1562, 2015) to interpolate commuting families <span>({S_{i}}_{i in mathcal {I}})</span> of contractions on a Hilbert space <span>(mathcal {H})</span>, to commuting families <span>({T_{i}}_{i in mathcal {I}})</span> of contractive <span>(mathcal {C}_{0})</span>-semigroups on <span>(L^{2}(prod _{i in mathcal {I}}mathbb {T}) otimes mathcal {H})</span>. As an excursus, we provide applications of the interpolations to time-discretisation and the embedding problem. Applied to Parrott’s construction (1970), we then demonstrate for <span>(d in mathbb {N})</span> with <span>(d ge 3)</span> the existence of commuting families <span>({T_{i}}_{i=1}^{d})</span> of contractive <span>(mathcal {C}_{0})</span>-semigroups which admit no simultaneous unitary dilation. As an application of these counter-examples, we obtain the residuality wrt.the topology of uniform <span>(textsc {wot})</span>-convergence on compact subsets of <span>(mathbb {R}_{ge 0}^{d})</span> of non-unitarily dilatable and non-unitarily approximable <i>d</i>-parameter contractive <span>(mathcal {C}_{0})</span>-semigroups on separable infinite-dimensional Hilbert spaces for each <span>(d ge 3)</span>. Similar results are also developed for <i>d</i>-tuples of commuting contractions. And by building on the counter-examples of Varopoulos-Kaijser (1973–74), a 0-1-result is obtained for the von Neumann inequality. Finally, we discuss applications to rigidity as well as the embedding problem, viz. that ‘typical’ pairs of commuting operators can be simultaneously embedded into commuting pairs of <span>(mathcal {C}_{0})</span>-semigroups, which extends results of Eisner (2009–2010).</p>","PeriodicalId":55400,"journal":{"name":"Banach Journal of Mathematical Analysis","volume":"302 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2024-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140610273","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
The finite Jung constant in Banach spaces 巴拿赫空间中的有限荣格常数
IF 1.2 2区 数学
Banach Journal of Mathematical Analysis Pub Date : 2024-04-15 DOI: 10.1007/s43037-024-00341-1
Jesús M. F. Castillo, Pier Luigi Papini
{"title":"The finite Jung constant in Banach spaces","authors":"Jesús M. F. Castillo, Pier Luigi Papini","doi":"10.1007/s43037-024-00341-1","DOIUrl":"https://doi.org/10.1007/s43037-024-00341-1","url":null,"abstract":"<p>We study in this paper the finite Jung constant, its interplay with Kottman’s constant and its meaning regarding the geometry of Banach spaces.</p>","PeriodicalId":55400,"journal":{"name":"Banach Journal of Mathematical Analysis","volume":"22 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2024-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140596222","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Canonical embedding of Lipschitz-free p-spaces 无 Lipschitz p 空间的典型嵌入
IF 1.2 2区 数学
Banach Journal of Mathematical Analysis Pub Date : 2024-04-15 DOI: 10.1007/s43037-024-00339-9
Marek Cúth, Tomáš Raunig
{"title":"Canonical embedding of Lipschitz-free p-spaces","authors":"Marek Cúth, Tomáš Raunig","doi":"10.1007/s43037-024-00339-9","DOIUrl":"https://doi.org/10.1007/s43037-024-00339-9","url":null,"abstract":"<p>We find a new finite algorithm for evaluation of Lipschitz-free <i>p</i>-space norm in finite-dimensional Lipschitz-free <i>p</i>-spaces. We use this algorithm to deal with the problem of whether given <i>p</i>-metric spaces <span>(mathcal {N}subset mathcal {M},)</span> the canonical embedding of <span>(mathcal {F}_p(mathcal {N}))</span> into <span>(mathcal {F}_p(mathcal {M}))</span> is an isomorphism. The most significant result in this direction is that the answer is positive if <span>(mathcal {N}subset mathcal {M})</span> are metric spaces.</p>","PeriodicalId":55400,"journal":{"name":"Banach Journal of Mathematical Analysis","volume":"68 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2024-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140596370","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Fredholm complements of upper triangular operator matrices 上三角算子矩阵的弗雷德霍尔补集
IF 1.2 2区 数学
Banach Journal of Mathematical Analysis Pub Date : 2024-04-13 DOI: 10.1007/s43037-024-00340-2
Sinan Qiu, Lining Jiang
{"title":"Fredholm complements of upper triangular operator matrices","authors":"Sinan Qiu, Lining Jiang","doi":"10.1007/s43037-024-00340-2","DOIUrl":"https://doi.org/10.1007/s43037-024-00340-2","url":null,"abstract":"<p>For a given operator pair <span>((A,B)in (B(H),B(K)))</span>, we denote by <span>(M_C)</span> the operator acting on a complex infinite dimensional separable Hilbert space <span>(Hoplus K)</span> of the form <span>(M_C=bigl ( {begin{matrix} A&amp;{}C 0&amp;{}B end{matrix}}bigr ))</span>. This paper focuses on the Fredholm complement problems of <span>(M_C)</span>. Namely, via the operator pair (<i>A</i>, <i>B</i>), we look for an operator <span>(Cin B(K,H))</span> such that <span>(M_C)</span> is Fredholm of finite ascent with nonzero nullity. As an application, we initiate the concept of the property (<i>C</i>) as a variant of Weyl’s theorem. At last, the stability of property (<i>C</i>) for <span>(2times 2)</span> upper triangular operator matrices is investigated by the virtue of the so-called entanglement spectra of the operator pair (<i>A</i>, <i>B</i>).</p>","PeriodicalId":55400,"journal":{"name":"Banach Journal of Mathematical Analysis","volume":"16 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2024-04-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140596225","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Norm inequalities in $${mathcal {L}}({mathcal {X}})$$ and a geometric constant $${mathcal {L}}({mathcal {X}})$$ 中的规范不等式和一个几何常数
IF 1.2 2区 数学
Banach Journal of Mathematical Analysis Pub Date : 2024-04-13 DOI: 10.1007/s43037-024-00342-0
Pintu Bhunia, Arpita Mal
{"title":"Norm inequalities in $${mathcal {L}}({mathcal {X}})$$ and a geometric constant","authors":"Pintu Bhunia, Arpita Mal","doi":"10.1007/s43037-024-00342-0","DOIUrl":"https://doi.org/10.1007/s43037-024-00342-0","url":null,"abstract":"<p>We introduce a new norm (say <span>(alpha )</span>-norm) on <span>({mathcal {L}}({mathcal {X}}),)</span> the space of all bounded linear operators defined on a normed linear space <span>({mathcal {X}})</span>. We explore various properties of the <span>(alpha )</span>-norm. In addition, we study several equalities and inequalities of the <span>(alpha )</span>-norm of operators on <span>({mathcal {X}}.)</span> 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 <span>(alpha )</span>-norm of operators by using the extreme points of the unit ball of the corresponding spaces. Furthermore, we define a geometric constant (say <span>(alpha )</span>-index) associated with <span>({mathcal {X}})</span> and study properties of the <span>(alpha )</span>-index. In particular, we obtain the exact value of the <span>(alpha )</span>-index for some polyhedral spaces and complex Hilbert space. Finally, we study the <span>(alpha )</span>-index of <span>(ell _p)</span>-sum of normed linear spaces.</p>","PeriodicalId":55400,"journal":{"name":"Banach Journal of Mathematical Analysis","volume":"68 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2024-04-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140596366","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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