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Advances in Operator Theory最新文献

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Data approximation in twisted shift-invariant spaces 扭曲移变空间中的数据逼近
IF 0.8
Advances in Operator Theory Pub Date : 2024-04-11 DOI: 10.1007/s43036-024-00336-7
Radha Ramakrishnan, Rabeetha Velsamy
{"title":"Data approximation in twisted shift-invariant spaces","authors":"Radha Ramakrishnan,&nbsp;Rabeetha Velsamy","doi":"10.1007/s43036-024-00336-7","DOIUrl":"10.1007/s43036-024-00336-7","url":null,"abstract":"<div><p>Twisted convolution is a non-standard convolution which arises while transferring the convolution of the Heisenberg group to the complex plane. Under this operation of twisted convolution, <span>(L^{1}(mathbb {R}^{2n}))</span> turns out to be a non-commutative Banach algebra. Hence the study of (twisted) shift-invariant spaces on <span>(mathbb {R}^{2n})</span> completely differs from the perspective of the usual shift-invariant spaces on <span>(mathbb {R}^{d})</span>. In this paper, by considering a set of functional data <span>(mathcal {F}={f_{1},ldots ,f_{m}})</span> in <span>(L^{2}(mathbb {R}^{2n}))</span>, we construct a finitely generated twisted shift-invariant space <span>(V^{t})</span> on <span>(mathbb {R}^{2n})</span> in such a way that the corresponding system of twisted translates of generators form a Parseval frame sequence and show that it gives the best approximation for a given data, in the sense of least square error. We also find the error of approximation of <span>(mathcal {F})</span> by <span>(V^{t})</span>. Finally, we illustrate this theory with an example.</p></div>","PeriodicalId":44371,"journal":{"name":"Advances in Operator Theory","volume":"9 2","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140792954","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Correction: Relating moments of self-adjoint polynomials in two orthogonal projections 更正：两个正交投影中自相关多项式的相关矩
IF 0.8
Advances in Operator Theory Pub Date : 2024-04-10 DOI: 10.1007/s43036-024-00343-8
Nizar Demni, Tarek Hamdi
{"title":"Correction: Relating moments of self-adjoint polynomials in two orthogonal projections","authors":"Nizar Demni,&nbsp;Tarek Hamdi","doi":"10.1007/s43036-024-00343-8","DOIUrl":"10.1007/s43036-024-00343-8","url":null,"abstract":"","PeriodicalId":44371,"journal":{"name":"Advances in Operator Theory","volume":"9 2","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-04-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140790033","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On (^*)-fusion frames for Hilbert (C^*)-modules 论希尔伯特 $$C^*$$ 模块的 $$^*$$ 融合框架
IF 0.8
Advances in Operator Theory Pub Date : 2024-04-10 DOI: 10.1007/s43036-024-00337-6
Nadia Assila, Samir Kabbaj, Hicham Zoubeir
{"title":"On (^*)-fusion frames for Hilbert (C^*)-modules","authors":"Nadia Assila,&nbsp;Samir Kabbaj,&nbsp;Hicham Zoubeir","doi":"10.1007/s43036-024-00337-6","DOIUrl":"10.1007/s43036-024-00337-6","url":null,"abstract":"<div><p>Our paper aims to extend fusion frames to Hilbert C<span>(^{*})</span>-modules. We introduce <span>(^*)</span>-fusion frames associated to weighted sequences of closed orthogonally complemented submodules, showcasing similarities to Hilbert space frames. Using Dragan S. Djordjevic’s distance, we define submodule angles and establish a new topology on the set of sequences of closed orthogonally complemented submodules. Relying on this topology, we obtain for our <span>(^*)</span>-fusion frames, some new perturbation results of topological and geometric character.</p></div>","PeriodicalId":44371,"journal":{"name":"Advances in Operator Theory","volume":"9 2","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-04-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140771409","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Free random variables whose free distributions are dictated by the semicircular law 自由分布由半圆律决定的自由随机变量
IF 0.8
Advances in Operator Theory Pub Date : 2024-04-09 DOI: 10.1007/s43036-024-00334-9
Ilwoo Cho, Palle E. T. Jorgensen
{"title":"Free random variables whose free distributions are dictated by the semicircular law","authors":"Ilwoo Cho,&nbsp;Palle E. T. Jorgensen","doi":"10.1007/s43036-024-00334-9","DOIUrl":"10.1007/s43036-024-00334-9","url":null,"abstract":"<div><p>In this paper, we study free-probabilistic structures of <span>( C^{*} )</span>-algebras generated by mutually free, multi free random variables followed by the semicircular law in a certain sense. Our main results (i) show that a semicircular element in a <span>( C^{*} )</span>-probability space <span>(left( A,varphi right) )</span>, and a certain <span>( * )</span>-isomorphism on <i>A</i> generate countable-infinitely many free random variables followed by the semicircular law, (ii) illustrate that, from mutually free, multi free random variables of (i), the corresponding <span>( C^{*} )</span>-probability spaces are well-determined, and (iii) characterize not only free-probabilistic structures, but also free-distributional data on the <span>( C^{*} )</span>-probability spaces of (ii). As application, we study some types of free random variables followed by the circular law, and followed by free Poisson distributions in certain senses.</p></div>","PeriodicalId":44371,"journal":{"name":"Advances in Operator Theory","volume":"9 2","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140759192","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On the Carathéodory–Schur interpolation problem over quaternions 关于四元数上的卡拉瑟奥多里-舒尔插值问题
IF 0.8
Advances in Operator Theory Pub Date : 2024-04-03 DOI: 10.1007/s43036-024-00329-6
Vladimir Bolotnikov
{"title":"On the Carathéodory–Schur interpolation problem over quaternions","authors":"Vladimir Bolotnikov","doi":"10.1007/s43036-024-00329-6","DOIUrl":"10.1007/s43036-024-00329-6","url":null,"abstract":"<div><p>We consider the quaternion version of the Toeplitz matrix extension problem with prescribed number of negative eigenvalues. The positive semidefinite case is closely related to the Carathéodory–Schur interpolation problem (<b>CSP</b>) in the Schur class <span>(mathcal S_{{mathbb {H}}})</span> and the Carathéodory class <span>({mathcal {C}}_{{mathbb {H}}})</span> of slice-regular functions on the unit quaternionic ball <span>({mathbb {B}})</span> that are, respectively, bounded by one in modulus and having positive real part in <span>({mathbb {B}})</span>. Explicit linear fractional parametrization formulas with free Schur-class parameter for the solution set of the <b>CSP</b> (in the indeterminate case) are given. Carathéodory–Fejér extremal problem and Carathéodory theorem on uniform approximation of a Schur-class function by quaternion finite Blaschke products are also derived. The indefinite version of the Toeplitz extension problem is applied to solve the <b>CSP</b> in the quaternion generalized Schur class. The linear fractional parametrization of the solution set for the indefinite indeterminate problem still exists, but some parameters should be excluded. These excluded parameters and the corresponding “quasi-solutions\" are classified and discussed in detail.</p></div>","PeriodicalId":44371,"journal":{"name":"Advances in Operator Theory","volume":"9 2","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140756432","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Truncations of operators in ({mathcal {B}}({mathcal {H}})) and their preservers $${mathcal {B}}({mathcal {H}})$$中算子的截断及其保值器
IF 0.8
Advances in Operator Theory Pub Date : 2024-04-03 DOI: 10.1007/s43036-024-00332-x
Yanling Mao, Guoxing Ji
{"title":"Truncations of operators in ({mathcal {B}}({mathcal {H}})) and their preservers","authors":"Yanling Mao,&nbsp;Guoxing Ji","doi":"10.1007/s43036-024-00332-x","DOIUrl":"10.1007/s43036-024-00332-x","url":null,"abstract":"<div><p>Let <span>(mathcal {H})</span> be a complex Hilbert space with <span>(dim {mathcal {H}}ge 2)</span> and <span>(mathcal {B}(mathcal {H}))</span> be the algebra of all bounded linear operators on <span>(mathcal {H})</span>. For <span>(A, B in mathcal {B}(mathcal {H}))</span>, <i>B</i> is called a truncation of <i>A</i>, denoted by <span>(Bprec A)</span>, if <span>(B=PAQ)</span> for some projections <span>(P,Qin {mathcal {B}}({mathcal {H}}))</span>. And <i>B</i> is called a maximal truncation of <i>A</i> if <span>(Bnot =A)</span> and there is no other truncation <i>C</i> of <i>A</i> such that <span>(Bprec C)</span>. We give necessary and sufficient conditions for <i>B</i> to be a maximal truncation of <i>A</i>. Using these characterizations, we determine structures of all bijections preserving truncations of operators in both directions on <span>(mathcal {B}(mathcal {H}))</span>.</p></div>","PeriodicalId":44371,"journal":{"name":"Advances in Operator Theory","volume":"9 2","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140760160","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Fejér–Riesz factorization in the QRC-subalgebra and circularity of the quaternionic numerical range QRC 子代数中的 Fejér-Riesz 因式分解和四元数程的圆周性
IF 0.8
Advances in Operator Theory Pub Date : 2024-04-03 DOI: 10.1007/s43036-024-00330-z
Alma van der Merwe, Madelein van Straaten, Hugo J. Woerdeman
{"title":"Fejér–Riesz factorization in the QRC-subalgebra and circularity of the quaternionic numerical range","authors":"Alma van der Merwe,&nbsp;Madelein van Straaten,&nbsp;Hugo J. Woerdeman","doi":"10.1007/s43036-024-00330-z","DOIUrl":"10.1007/s43036-024-00330-z","url":null,"abstract":"<div><p>We provide a characterization when the quaternionic numerical range of a matrix is a closed ball with center 0. The proof makes use of Fejér–Riesz factorization of matrix-valued trigonometric polynomials within the algebra of complex matrices associated with quaternion matrices.</p></div>","PeriodicalId":44371,"journal":{"name":"Advances in Operator Theory","volume":"9 2","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s43036-024-00330-z.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140783695","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Frame dimension functions and phase retrievability 帧维函数和相位检索
IF 0.8
Advances in Operator Theory Pub Date : 2024-04-03 DOI: 10.1007/s43036-024-00331-y
Deguang Han, Kai Liu
{"title":"Frame dimension functions and phase retrievability","authors":"Deguang Han,&nbsp;Kai Liu","doi":"10.1007/s43036-024-00331-y","DOIUrl":"10.1007/s43036-024-00331-y","url":null,"abstract":"<div><p>The frame dimension function of a frame <span>({{mathcal {F}}}= {f_j}_{j=1}^{n})</span> for an <i>n</i>-dimensional Hilbert space <i>H</i> is the function <span>(d_{{{mathcal {F}}}}(x) = dim {textrm{span}}{ langle x, f_{j}rangle f_{j}: j=1,ldots , N}, 0ne xin H.)</span> It is known that <span>({{mathcal {F}}})</span> does phase retrieval for an <i>n</i>-dimensional real Hilbert space <i>H</i> if and only if <span>({textrm{range}} (d_{{{mathcal {F}}}}) = { n}.)</span> This indicates that the range of the dimension function is one of the good candidates to measure the phase retrievability for an arbitrary frame. In this paper we investigate some structural properties for the range of the dimension function, and examine the connections among different exactness of a frame with respect to its PR-redundance, dimension function and range of the dimension function. A subset <span>(Omega )</span> of <span>({1,ldots , n})</span> containing <i>n</i> is attainable if <span>({textrm{range}} (d_{{{mathcal {F}}}}) = Omega )</span> for some frame <span>({{mathcal {F}}}.)</span> With the help of linearly connected frames, we show that, while not every <span>(Omega )</span> is attainable, every (integer) interval containing <i>n</i> is always attainable by an <i>n</i>-linearly independent frame. Consequently, <span>({textrm{range}}(d_{{{mathcal {F}}}}))</span> is an interval for every generic frame for <span>({mathbb {R}},^n.)</span> Additionally, we also discuss and post some questions related to the connections among ranges of the dimension functions, linearly connected frames and maximal phase retrievable subspaces.</p></div>","PeriodicalId":44371,"journal":{"name":"Advances in Operator Theory","volume":"9 2","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140771161","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Numerical radius and geometric means of real power 实际功率的数值半径和几何平均数
IF 0.8
Advances in Operator Theory Pub Date : 2024-03-25 DOI: 10.1007/s43036-024-00328-7
Yuki Seo
{"title":"Numerical radius and geometric means of real power","authors":"Yuki Seo","doi":"10.1007/s43036-024-00328-7","DOIUrl":"10.1007/s43036-024-00328-7","url":null,"abstract":"<div><p>Norm inequalities related to geometric means are discussed by many researchers. Though the operator norm is unitarily invariant one, the numerical radius is not so and unitarily similar. In this paper, we prove some numerical radius inequalities that are related to operator geometric means and spectral geometric ones of real power for positive invertible operators.</p></div>","PeriodicalId":44371,"journal":{"name":"Advances in Operator Theory","volume":"9 2","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140384228","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On the geometry of an order unit space 关于阶单元空间的几何
IF 0.8
Advances in Operator Theory Pub Date : 2024-03-24 DOI: 10.1007/s43036-024-00327-8
Anil Kumar Karn
{"title":"On the geometry of an order unit space","authors":"Anil Kumar Karn","doi":"10.1007/s43036-024-00327-8","DOIUrl":"10.1007/s43036-024-00327-8","url":null,"abstract":"<div><p>We introduce the notion of <i>skeleton</i> with a head in a non-zero real vector space. We prove that skeletons with a head describe order unit spaces geometrically. Next, we prove that the skeleton consists of boundary elements of the positive cone of norm one. We discuss some elementary properties of the skeleton. We also find a condition under which <i>V</i> contains a copy of <span>(ell _{infty }^n)</span> for some <span>(n in {mathbb {N}})</span> as an order unit subspace.</p></div>","PeriodicalId":44371,"journal":{"name":"Advances in Operator Theory","volume":"9 2","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-03-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142413544","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","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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