{"title":"Numerical Radius Inequalities for Sums and Products of Operators","authors":"Wasim Audeh","doi":"10.4236/ALAMT.2019.93003","DOIUrl":"https://doi.org/10.4236/ALAMT.2019.93003","url":null,"abstract":"A numerical radius inequality due to Shebrawi and Albadawi says that: If Ai, Bi, Xi are bounded operators in Hilbert space, i = 1,2,..., n , and f,g be nonnegative continuous functions on [0, ∞) satisfying the relation f(t)g(t) = t (t∈[0, ∞)), then for all r≥1. We give sharper numerical radius inequality which states that: If Ai, Bi, Xi are bounded operators in Hilbert space, i = 1,2,..., n , and f,g be nonnegative continuous functions on [0, ∞) satisfying the relation f(t)g(t) = t (t∈[0, ∞)), then where . Moreover, we give many numerical radius inequalities which are sharper than related inequalities proved recently, and several applications are given.","PeriodicalId":65610,"journal":{"name":"线性代数与矩阵理论研究进展(英文)","volume":"19 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83475427","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}
{"title":"Bounding Inequalities for Eigenvalues of Principal Submatrices","authors":"A. Dax","doi":"10.4236/ALAMT.2019.92002","DOIUrl":"https://doi.org/10.4236/ALAMT.2019.92002","url":null,"abstract":"Ky Fan trace theorems and the interlacing theorems of Cauchy and Poincare are important observations that characterize Hermitian matrices. In this note, we introduce a new type of inequalities which extend these theorems. The new inequalities are obtained from the old ones by replacing eigenvalues and diagonal entries with their moduli. This modification yields effective bounding inequalities which are valid on a larger range of matrices.","PeriodicalId":65610,"journal":{"name":"线性代数与矩阵理论研究进展(英文)","volume":"2 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81528072","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}
{"title":"A Follow-Up on Projection Theory: Theorems and Group Action","authors":"Jean-Francois Niglio","doi":"10.4236/ALAMT.2019.91001","DOIUrl":"https://doi.org/10.4236/ALAMT.2019.91001","url":null,"abstract":"In this article, we wish to expand on some of the results obtained from the first article entitled Projection Theory. We have already established that one-parameter projection operators can be constructed from the unit circle . As discussed in the previous article these operators form a Lie group known as the Projection Group. In the first section, we will show that the concepts from my first article are consistent with existing theory [1] [2]. In the second section, it will be demonstrated that not only such operators are mutually congruent but also we can define a group action on by using the rotation group [3] [4]. It will be proved that the group acts on elements of in a non-faithful but ∞-transitive way consistent with both group operations. Finally, in the last section we define the group operation in terms of matrix operations using the operator and the Hadamard Product; this construction is consistent with the group operation defined in the first article.","PeriodicalId":65610,"journal":{"name":"线性代数与矩阵理论研究进展(英文)","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-03-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85502910","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}
{"title":"A Minimality of the Rational Canonical Form","authors":"H. Liu, Honglian Zhang","doi":"10.4236/alamt.2019.94006","DOIUrl":"https://doi.org/10.4236/alamt.2019.94006","url":null,"abstract":"The rational canonical form theorem is very essential basic result of matrix theory, which has been proved by different methods in the literature. In this note, we provide an effcient direct proof, from which the minimality for the decomposition of the rational canonical form can be found.","PeriodicalId":65610,"journal":{"name":"线性代数与矩阵理论研究进展(英文)","volume":"57 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80432754","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}
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