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Algorithm for spectral factorization of polynomial matrices on the real line 实线上多项式矩阵的谱因式分解算法
IF 0.8
Advances in Operator Theory Pub Date : 2024-11-29 DOI: 10.1007/s43036-024-00406-w
Lasha Ephremidze
{"title":"Algorithm for spectral factorization of polynomial matrices on the real line","authors":"Lasha Ephremidze","doi":"10.1007/s43036-024-00406-w","DOIUrl":"10.1007/s43036-024-00406-w","url":null,"abstract":"<div><p>In this paper, we extend the basic idea of the Janashia–Lagvilava algorithm to adapt it for the spectral factorization of positive-definite polynomial matrices on the real line. This extension results in a new spectral factorization algorithm for polynomial matrix functions defined on <span>(mathbb {R})</span>. The presented numerical example demonstrates that the proposed algorithm outperforms an existing algorithm in terms of accuracy.</p></div>","PeriodicalId":44371,"journal":{"name":"Advances in Operator Theory","volume":"10 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142737038","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
Stability in non-normal periodic Jacobi operators: advancing Börg’s theorem 非正态周期雅可比算子的稳定性：推进伯格定理
IF 0.8
Advances in Operator Theory Pub Date : 2024-11-28 DOI: 10.1007/s43036-024-00402-0
G. Krishna Kumar, V. B. Kiran Kumar
{"title":"Stability in non-normal periodic Jacobi operators: advancing Börg’s theorem","authors":"G. Krishna Kumar,&nbsp;V. B. Kiran Kumar","doi":"10.1007/s43036-024-00402-0","DOIUrl":"10.1007/s43036-024-00402-0","url":null,"abstract":"<div><p>Periodic Jacobi operators naturally arise in numerous applications, forming a cornerstone in various fields. The spectral theory associated with these operators boasts an extensive body of literature. Considered as discretized counterparts of Schrödinger operators, widely employed in quantum mechanics, Jacobi operators play a crucial role in mathematical formulations. The classical uniqueness result by G. Börg in 1946 occupies a significant place in the literature of inverse spectral theory and its applications. This result is closely intertwined with M. Kac’s renowned article, ‘Can one hear the shape of a drum?’ published in 1966. Since 1975,  discrete versions of Börg’s theorem have been available in the literature. In this article, we concentrate on the non-normal periodic Jacobi operator and the discrete versions of Börg’s Theorem. We extend recently obtained stability results to cover non-normal cases. The existing stability findings establish a correlation between the oscillations of the matrix entries and the size of the spectral gap. Our result covers the current self-adjoint versions of Börg’s theorem, including recent quantitative variations. Here, the oscillations of the matrix entries are linked to the path-connectedness of the pseudospectrum. Additionally, we explore finite difference approximations of various linear differential equations as specific applications.</p></div>","PeriodicalId":44371,"journal":{"name":"Advances in Operator Theory","volume":"10 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142736900","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 maximal hyperplane sections of the unit ball of (l_p^n) for (p>2) 关于（p>2）的（l_p^n）单位球的最大超平面部分
IF 0.8
Advances in Operator Theory Pub Date : 2024-11-21 DOI: 10.1007/s43036-024-00404-y
Hermann König
{"title":"On maximal hyperplane sections of the unit ball of (l_p^n) for (p>2)","authors":"Hermann König","doi":"10.1007/s43036-024-00404-y","DOIUrl":"10.1007/s43036-024-00404-y","url":null,"abstract":"<div><p>The maximal hyperplane section of the <span>(l_infty ^n)</span>-ball, i.e. of the <i>n</i>-cube, is the one perpendicular to <span>(frac{1}{sqrt{2}} (1,1,0 ,ldots ,0))</span>, as shown by Ball. Eskenazis, Nayar and Tkocz extended this result to the <span>(l_p^n)</span>-balls for very large <span>(p ge 10^{15})</span>. By Oleszkiewicz, Ball’s result does not transfer to <span>(l_p^n)</span> for <span>(2&lt; p &lt; p_0 simeq 26.265)</span>. Then the hyperplane section perpendicular to the main diagonal yields a counterexample for large dimensions <i>n</i>. Suppose that <span>(p_0 le p &lt; infty )</span>. We show that the analogue of Ball’s result holds in <span>(l_p^n)</span>-balls for all hyperplanes with normal unit vectors <i>a</i>, if all coordinates of <i>a</i> have modulus <span>(le frac{1}{sqrt{2}})</span> and <i>p</i> has distance <span>(ge 2^{-p})</span> to the even integers. Under similar assumptions, we give a Gaussian upper bound for <span>(20&lt; p &lt; p_0)</span>.</p></div>","PeriodicalId":44371,"journal":{"name":"Advances in Operator Theory","volume":"10 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-11-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s43036-024-00404-y.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142679823","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
Commutativity and spectral properties for a general class of Szász–Mirakjan–Durrmeyer operators 一般类 Szász-Mirakjan-Durrmeyer 算子的交换性和谱特性
IF 0.8
Advances in Operator Theory Pub Date : 2024-11-21 DOI: 10.1007/s43036-024-00403-z
Ulrich Abel, Ana Maria Acu, Margareta Heilmann, Ioan Raşa
{"title":"Commutativity and spectral properties for a general class of Szász–Mirakjan–Durrmeyer operators","authors":"Ulrich Abel,&nbsp;Ana Maria Acu,&nbsp;Margareta Heilmann,&nbsp;Ioan Raşa","doi":"10.1007/s43036-024-00403-z","DOIUrl":"10.1007/s43036-024-00403-z","url":null,"abstract":"<div><p>In this paper we present commutativity results for a general class of Szász–Mirakjan–Durrmeyer type operators and associated differential operators and investigate their eigenfunctions.Please confirm if the inserted city names are correct. Amend if necessary.The inserted city name is correct.</p></div>","PeriodicalId":44371,"journal":{"name":"Advances in Operator Theory","volume":"10 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-11-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s43036-024-00403-z.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142679819","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
Matrices with hyperbolical Krein space numerical range 具有双曲克雷因空间数值范围的矩阵
IF 0.8
Advances in Operator Theory Pub Date : 2024-11-12 DOI: 10.1007/s43036-024-00399-6
N. Bebiano, R. Lemos, G. Soares
{"title":"Matrices with hyperbolical Krein space numerical range","authors":"N. Bebiano,&nbsp;R. Lemos,&nbsp;G. Soares","doi":"10.1007/s43036-024-00399-6","DOIUrl":"10.1007/s43036-024-00399-6","url":null,"abstract":"<div><p>This paper is devoted to matrices with hyperbolical Krein space numerical range. This shape characterizes the 2-by-2 case and persists for certain classes of matrices, independently of their size. Necessary and sufficient conditions for low dimensional tridiagonal matrices to have this shape are obtained involving only the matrix entries.</p></div>","PeriodicalId":44371,"journal":{"name":"Advances in Operator Theory","volume":"10 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-11-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s43036-024-00399-6.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142600626","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
On the (m, n)-clock problem and the (ell _{infty }-ell _1) norm of a matrix 关于（m, n）-时钟问题和矩阵的（ell _{infty }-ell _1）规范
IF 0.8
Advances in Operator Theory Pub Date : 2024-11-12 DOI: 10.1007/s43036-024-00401-1
Chandrodoy Chattopadhyay, Kalidas Mandal, Debmalya Sain
{"title":"On the (m, n)-clock problem and the (ell _{infty }-ell _1) norm of a matrix","authors":"Chandrodoy Chattopadhyay,&nbsp;Kalidas Mandal,&nbsp;Debmalya Sain","doi":"10.1007/s43036-024-00401-1","DOIUrl":"10.1007/s43036-024-00401-1","url":null,"abstract":"<div><p>We characterize the norm attainment set of a linear operator from <span>( ell _{infty }^{2}({mathbb {C}}) )</span> to <span>( ell _{1}^{2}({mathbb {C}}), )</span> with the help of a physical model involving two clocks entangled in a specific way. More generally, we introduce the (<i>m</i>, <i>n</i>)-clock Problem and establish its equivalence with computing the <span>(ell _{infty }-ell _1)</span> norm of an <span>( m times n )</span> matrix. We further give an explicit description of the smooth and the non-smooth points in <span>({mathbb {L}}big (ell _infty ^2({mathbb {C}}),ell _1^2({mathbb {C}})big ).)</span></p></div>","PeriodicalId":44371,"journal":{"name":"Advances in Operator Theory","volume":"10 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-11-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142600627","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
Some singular value inequalities on commutators 关于换元的一些奇异值不等式
IF 0.8
Advances in Operator Theory Pub Date : 2024-11-09 DOI: 10.1007/s43036-024-00393-y
Maninderjit Kaur, Isha Garg
{"title":"Some singular value inequalities on commutators","authors":"Maninderjit Kaur,&nbsp;Isha Garg","doi":"10.1007/s43036-024-00393-y","DOIUrl":"10.1007/s43036-024-00393-y","url":null,"abstract":"<div><p>In this study, singular value and norm inequalities for expressions of the form <span>(SXT+Y)</span> are established. It is shown that if <span>(S,T,X,Y in mathcal {B(H)})</span> such that <i>X</i>, <i>Y</i> are compact operators, then </p><div><div><span>$$begin{aligned} sigma _{j}left( SXT+Yright) le left( Vert SVert Vert TVert + Vert YVert right) sigma _j( Xoplus I).end{aligned}$$</span></div></div><p>Additionally, we explore several applications of this inequality, which provide a broader framework for analysis and yield more nuanced insights. For <span>(X, Yin mathcal {B(H)})</span> one notable application is the following inequality, </p><div><div><span>$$begin{aligned} sigma _{j}left( mid X-Ymid ^{2}-2 left( mid X mid ^{2}+mid Y mid ^{2} right) right) le left( 1+mid mid Ymid mid right) ^{2} sigma _{j}( mid X mid ^{2}oplus I). end{aligned}$$</span></div></div><p>These results extend existing inequalities and offer new perspectives in operator theory.</p></div>","PeriodicalId":44371,"journal":{"name":"Advances in Operator Theory","volume":"10 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142598904","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
Dominated and absolutely summing operators on the space (,C_{rc}(X,E)) of vector-valued continuous functions 矢量连续函数空间 （,C_{rc}(X,E)）上的支配和绝对求和算子
IF 0.8
Advances in Operator Theory Pub Date : 2024-11-05 DOI: 10.1007/s43036-024-00398-7
Marian Nowak
{"title":"Dominated and absolutely summing operators on the space (,C_{rc}(X,E)) of vector-valued continuous functions","authors":"Marian Nowak","doi":"10.1007/s43036-024-00398-7","DOIUrl":"10.1007/s43036-024-00398-7","url":null,"abstract":"<div><p>Let <i>X</i> be a completely regular Hausdorff space and <i>E</i> and <i>F</i> be Banach spaces. Let <span>(C_{rc}(X,E))</span> denote the Banach space of all continuous functions <span>(f:Xrightarrow E)</span> such that <i>f</i>(<i>X</i>) is a relatively compact set in <i>E</i>, and <span>(beta _sigma )</span> be the strict topology on <span>(C_{rc}(X,E))</span>. We characterize dominated and absolutely summing operators <span>(T:C_{rc}(X,E)rightarrow F)</span> in terms of their representing operator-valued Baire measures. It is shown that every absolutely summing <span>((beta _sigma ,Vert cdot Vert _F))</span>-continuous operator <span>(T:C_{rc}(X,E)rightarrow F)</span> is dominated. Moreover, we obtain that every dominated operator <span>(T:C_{rc}(X,E)rightarrow F)</span> is absolutely summing if and only if every bounded linear operator <span>(U:Erightarrow F)</span> is absolutely summing.</p></div>","PeriodicalId":44371,"journal":{"name":"Advances in Operator Theory","volume":"10 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s43036-024-00398-7.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142587775","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
Representation and inequalities involving continuous linear functionals and fractional derivatives 涉及连续线性函数和分数导数的表示法和不等式
IF 0.8
Advances in Operator Theory Pub Date : 2024-10-29 DOI: 10.1007/s43036-024-00397-8
Marc Jornet, Juan J. Nieto
{"title":"Representation and inequalities involving continuous linear functionals and fractional derivatives","authors":"Marc Jornet,&nbsp;Juan J. Nieto","doi":"10.1007/s43036-024-00397-8","DOIUrl":"10.1007/s43036-024-00397-8","url":null,"abstract":"<div><p>We investigate how continuous linear functionals can be represented in terms of generic operators and certain kernels (Peano kernels), and we study lower bounds for the operators as a consequence, in the space of square-integrable functions. We apply and develop the theory for the Riemann–Liouville fractional derivative (an inverse of the Riemann–Liouville integral), where inequalities are derived with the Gaussian hypergeometric function. This work is inspired by the recent contributions by Fernandez and Buranay (J Comput Appl Math 441:115705, 2024) and Jornet (Arch Math, 2024).</p></div>","PeriodicalId":44371,"journal":{"name":"Advances in Operator Theory","volume":"10 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-10-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s43036-024-00397-8.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142540697","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
Operator product states on tensor powers of (C^*)-algebras 关于张量幂的（C^*）代数的算子乘积状态
IF 0.8
Advances in Operator Theory Pub Date : 2024-10-28 DOI: 10.1007/s43036-024-00389-8
Emil Prodan
{"title":"Operator product states on tensor powers of (C^*)-algebras","authors":"Emil Prodan","doi":"10.1007/s43036-024-00389-8","DOIUrl":"10.1007/s43036-024-00389-8","url":null,"abstract":"<div><p>The program of matrix product states on tensor powers <span>({mathcal {A}}^{otimes {mathbb {Z}}})</span> of <span>(C^*)</span>-algebras is carried under the assumption that <span>({mathcal {A}})</span> is an arbitrary nuclear C*-algebra. For any shift invariant state <span>(omega )</span>, we demonstrate the existence of an order kernel ideal <span>({mathcal {K}}_omega )</span>, whose quotient action reduces and factorizes the initial data <span>(({mathcal {A}}^{otimes {mathbb {Z}}}, omega ))</span> to the tuple <span>(({mathcal {A}},{mathcal {B}}_omega = {mathcal {A}}^{otimes {mathbb {N}}^times }/{mathcal {K}}_omega , {mathbb {E}}_omega : text{AA }otimes {mathcal {B}}_omega rightarrow {mathcal {B}}_omega , {bar{omega }}: {mathcal {B}}_omega rightarrow {mathbb {C}}))</span>, where <span>({mathcal {B}}_omega )</span> is an operator system and <span>({mathbb {E}}_omega )</span> and <span>({bar{omega }})</span> are unital and completely positive maps. Reciprocally, given a (input) tuple <span>(({mathcal {A}},{mathcal {S}},{mathbb {E}},phi ))</span> that shares similar attributes, we supply an algorithm that produces a shift-invariant state on <span>({mathcal {A}}^{otimes {mathbb {Z}}})</span>. We give sufficient conditions in which the so constructed states are ergodic and they reduce back to their input data. As examples, we formulate the input data that produces AKLT-type states, this time in the context of infinite dimensional site algebras <span>({mathcal {A}})</span>, such as the <span>(C^*)</span>-algebras of discrete amenable groups.</p></div>","PeriodicalId":44371,"journal":{"name":"Advances in Operator Theory","volume":"10 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-10-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142524377","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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