In this article, the relationship between Birkhoff-James orthogonality of elementary tensors in certain tensor product spaces with the Birkhoff-James orthogonality of individual elements in their respective spaces is studied.
{"title":"Birkhoff-James orthogonality in certain tensor products of Banach spaces","authors":"Mohit, R. Jain","doi":"10.7153/oam-2023-17-17","DOIUrl":"https://doi.org/10.7153/oam-2023-17-17","url":null,"abstract":"In this article, the relationship between Birkhoff-James orthogonality of elementary tensors in certain tensor product spaces with the Birkhoff-James orthogonality of individual elements in their respective spaces is studied.","PeriodicalId":56274,"journal":{"name":"Operators and Matrices","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-10-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45863838","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this article we study analogues of the weak expectation property of discrete group C*-algebras and their crossed products, in the discrete quantum group setting, i.e., discrete quantum group C*-algebras and crossed products of C*-algebras with amenable discrete quantum groups.
{"title":"Weak expectations of discrete quantum group algebras and crossed products","authors":"A. Bhattacharjee, Angshuman Bhattacharya","doi":"10.7153/oam-2023-17-06","DOIUrl":"https://doi.org/10.7153/oam-2023-17-06","url":null,"abstract":"In this article we study analogues of the weak expectation property of discrete group C*-algebras and their crossed products, in the discrete quantum group setting, i.e., discrete quantum group C*-algebras and crossed products of C*-algebras with amenable discrete quantum groups.","PeriodicalId":56274,"journal":{"name":"Operators and Matrices","volume":"1 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41361835","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The matrix S = [1 + x i y j ] ni,j =1 , 0 < x 1 < · · · < x n , 0 < y 1 < · · · < y n , has gained importance lately due to its role in powers preserving total nonnegativity. We give an explicit decomposition of S in terms of elementary bidiagonal matrices, which is analogous to the Neville decomposition . We give a bidiagonal decomposition of S ◦ m = [(1 + x i y j ) m ] for positive integers 1 ≤ m ≤ n − 1. We also explore the total positivity of Hadamard powers of another important class of matrices called mean matrices .
矩阵S=[1+x i y j]ni，j=1，0
{"title":"Bidiagonal decompositions and total positivity of some special matrices","authors":"Priyanka Grover, Veer Singh Panwar","doi":"10.7153/oam-2022-16-41","DOIUrl":"https://doi.org/10.7153/oam-2022-16-41","url":null,"abstract":"The matrix S = [1 + x i y j ] ni,j =1 , 0 < x 1 < · · · < x n , 0 < y 1 < · · · < y n , has gained importance lately due to its role in powers preserving total nonnegativity. We give an explicit decomposition of S in terms of elementary bidiagonal matrices, which is analogous to the Neville decomposition . We give a bidiagonal decomposition of S ◦ m = [(1 + x i y j ) m ] for positive integers 1 ≤ m ≤ n − 1. We also explore the total positivity of Hadamard powers of another important class of matrices called mean matrices .","PeriodicalId":56274,"journal":{"name":"Operators and Matrices","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49148398","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The main goal of this work is to present new matrix inequalities of the Cauchy-Schwarz type. In particular, we investigate the so-called Lieb functions, whose definition came as an umbrella of Cauchy-Schwarz-like inequalities, then we consider the mixed Cauchy-Schwarz inequality. This latter inequality has been influential in obtaining several other matrix inequalities, including numerical radius and norm results. Among many other results, we show that [left| T right|le frac{1}{4}left( left| left| T right|+left| {{T}^{*}} right|+2mathfrak RT right|+left| left| T right|+left| {{T}^{*}} right|-2mathfrak RT right| right),] where $mathfrak RT$ is the real part of $T$.
{"title":"On the matrix Cauchy-Schwarz inequality","authors":"M. Sababheh, C. Conde, H. Moradi","doi":"10.7153/oam-2023-17-34","DOIUrl":"https://doi.org/10.7153/oam-2023-17-34","url":null,"abstract":"The main goal of this work is to present new matrix inequalities of the Cauchy-Schwarz type. In particular, we investigate the so-called Lieb functions, whose definition came as an umbrella of Cauchy-Schwarz-like inequalities, then we consider the mixed Cauchy-Schwarz inequality. This latter inequality has been influential in obtaining several other matrix inequalities, including numerical radius and norm results. Among many other results, we show that [left| T right|le frac{1}{4}left( left| left| T right|+left| {{T}^{*}} right|+2mathfrak RT right|+left| left| T right|+left| {{T}^{*}} right|-2mathfrak RT right| right),] where $mathfrak RT$ is the real part of $T$.","PeriodicalId":56274,"journal":{"name":"Operators and Matrices","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42475478","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Square of a posinormal operator is not necessarily posinormal. But (i) powers of quasiposinormal operators are quasiposinormal and, under closed ranges assumption, powers of (ii) posinormal operators are posinormal, (iii) of operators that are both posinormal and coposinormal are posinormal and coposinormal, and (iv) of semi-Fredholm posinormal operators are posinormal.
{"title":"Powers of posinormal operators","authors":"C. Kubrusly, P. Vieira, J. Zanni","doi":"10.7153/OAM-10-02","DOIUrl":"https://doi.org/10.7153/OAM-10-02","url":null,"abstract":"Square of a posinormal operator is not necessarily posinormal. But (i) powers of quasiposinormal operators are quasiposinormal and, under closed ranges assumption, powers of (ii) posinormal operators are posinormal, (iii) of operators that are both posinormal and coposinormal are posinormal and coposinormal, and (iv) of semi-Fredholm posinormal operators are posinormal.","PeriodicalId":56274,"journal":{"name":"Operators and Matrices","volume":"1 1","pages":"15-27"},"PeriodicalIF":0.5,"publicationDate":"2022-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43064827","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
. Toeplitz matrices are ubiquitous and play important roles across many areas of mathematics. In this paper, we present some algebraic results concerning block Toeplitz matrices with block entries belonging to a commutative algebra A . The characterization of normal block Toeplitz matrices with entries from A is also obtained.
{"title":"On some algebraic properties of block Toeplitz matrices with commuting entries","authors":"M. A. Khan, A. Yagoub","doi":"10.7153/oam-2022-16-61","DOIUrl":"https://doi.org/10.7153/oam-2022-16-61","url":null,"abstract":". Toeplitz matrices are ubiquitous and play important roles across many areas of mathematics. In this paper, we present some algebraic results concerning block Toeplitz matrices with block entries belonging to a commutative algebra A . The characterization of normal block Toeplitz matrices with entries from A is also obtained.","PeriodicalId":56274,"journal":{"name":"Operators and Matrices","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-01-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46777201","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Adrian Fan, Jack Montemurro, P. Motakis, Naina Praveen, A. Rusonik, P. Skoufranis, N. Tobin
Motivated by an influential result of Bourgain and Tzafriri, we consider continuous matrix functions $A:mathbb{R}to M_{ntimes n}$ and lower $ell_2$-norm bounds associated with their restriction to certain subspaces. We prove that for any such $A$ with unit-length columns, there exists a continuous choice of subspaces $tmapsto U(t)subset mathbb{R}^n$ such that for $vin U(t)$, $|A(t)v|geq c|v|$ where $c$ is some universal constant. Furthermore, the $U(t)$ are chosen so that their dimension satisfies a lower bound with optimal asymptotic dependence on $n$ and $sup_{tin mathbb{R}}|A(t)|.$ We provide two methods. The first relies on an orthogonality argument, while the second is probabilistic and combinatorial in nature. The latter does not yield the optimal bound for $dim(U(t))$ but the $U(t)$ obtained in this way are guaranteed to have a canonical representation as joined-together spaces spanned by subsets of the unit vector basis.
受Bourgain和Tzafriri的一个有影响的结果的启发，我们考虑了连续矩阵函数$A:mathbb｛R｝到M_｛n times n｝$以及与它们对某些子空间的限制相关的下$ell_2$范数界。我们证明了对于任何这样的具有单位长度列的$A$，存在子空间$tmapsto U（t）subet mathbb｛R｝^n$的连续选择，使得对于U（t）$中的$v，$|A（t）v|geq c|v|$，其中$c$是某个通用常数。此外，选择$U（t）$使得它们的维数满足对$n$和$sup_｛tinmathbb｛R｝｝|a（t）|.$具有最优渐近依赖性的下界我们提供了两种方法。第一个依赖于正交性论证，而第二个本质上是概率性和组合性的。后者不产生$dim（U（t））$的最优界，但以这种方式获得的$U（t。
{"title":"Restricted invertibility of continuous matrix functions","authors":"Adrian Fan, Jack Montemurro, P. Motakis, Naina Praveen, A. Rusonik, P. Skoufranis, N. Tobin","doi":"10.7153/oam-2022-16-78","DOIUrl":"https://doi.org/10.7153/oam-2022-16-78","url":null,"abstract":"Motivated by an influential result of Bourgain and Tzafriri, we consider continuous matrix functions $A:mathbb{R}to M_{ntimes n}$ and lower $ell_2$-norm bounds associated with their restriction to certain subspaces. We prove that for any such $A$ with unit-length columns, there exists a continuous choice of subspaces $tmapsto U(t)subset mathbb{R}^n$ such that for $vin U(t)$, $|A(t)v|geq c|v|$ where $c$ is some universal constant. Furthermore, the $U(t)$ are chosen so that their dimension satisfies a lower bound with optimal asymptotic dependence on $n$ and $sup_{tin mathbb{R}}|A(t)|.$ We provide two methods. The first relies on an orthogonality argument, while the second is probabilistic and combinatorial in nature. The latter does not yield the optimal bound for $dim(U(t))$ but the $U(t)$ obtained in this way are guaranteed to have a canonical representation as joined-together spaces spanned by subsets of the unit vector basis.","PeriodicalId":56274,"journal":{"name":"Operators and Matrices","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-01-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46745343","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
. We consider an inverse problem for Dirac system in case where one of nonseparated boundary conditions involves a linear function of spectral parameter. We prove the uniqueness theorem for the solution of this problem and then, based on this theorem, we construct a solution algorithm for the considered problem.
{"title":"Solution algorithm of the inverse spectral problem for Dirac operator with a spectral parameter in the boundary condition","authors":"Abid G. Ferzullazadeh","doi":"10.7153/oam-2022-16-11","DOIUrl":"https://doi.org/10.7153/oam-2022-16-11","url":null,"abstract":". We consider an inverse problem for Dirac system in case where one of nonseparated boundary conditions involves a linear function of spectral parameter. We prove the uniqueness theorem for the solution of this problem and then, based on this theorem, we construct a solution algorithm for the considered problem.","PeriodicalId":56274,"journal":{"name":"Operators and Matrices","volume":"1 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71222856","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Determinantal polynomials of a weighted shift matrix with palindromic geometric weights","authors":"Undrakh Batzorig","doi":"10.7153/oam-2022-16-24","DOIUrl":"https://doi.org/10.7153/oam-2022-16-24","url":null,"abstract":"","PeriodicalId":56274,"journal":{"name":"Operators and Matrices","volume":"1 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71222973","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Generalizing the Ando-Hiai inequality for sectorial matrices","authors":"Linlong Zhao, Yanp ng Zheng, Xia yu Jiang","doi":"10.7153/oam-2022-16-26","DOIUrl":"https://doi.org/10.7153/oam-2022-16-26","url":null,"abstract":"","PeriodicalId":56274,"journal":{"name":"Operators and Matrices","volume":"1 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71223031","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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