Pub Date : 2019-11-05DOI: 10.7900/jot.2019feb26.2270
Alexandre Baldare, R. Come, M. Lesch, V. Nistor
Let Γ be a finite abelian group acting on a smooth, compact manifold M without boundary and let P∈ψm(M;E0,E1) be a Γ-invariant, classical, pseudodifferential operator acting between sections of two Γ-equivariant vector bundles. Let α be an irreducible representation of Γ. We obtain necessary and sufficient conditions for the restriction πα(P):Hs(M;E0)α→Hs−m(M;E1)α of P between the α-isotypical components of Sobolev spaces to be Fredholm.
{"title":"Fredholm conditions for invariant operators: finite abelian groups and boundary value problems","authors":"Alexandre Baldare, R. Come, M. Lesch, V. Nistor","doi":"10.7900/jot.2019feb26.2270","DOIUrl":"https://doi.org/10.7900/jot.2019feb26.2270","url":null,"abstract":"Let Γ be a finite abelian group acting on a smooth, compact manifold M without boundary and let P∈ψm(M;E0,E1) be a Γ-invariant, classical, pseudodifferential operator acting between sections of two Γ-equivariant vector bundles. Let α be an irreducible representation of Γ. We obtain necessary and sufficient conditions for the restriction πα(P):Hs(M;E0)α→Hs−m(M;E1)α of P between the α-isotypical components of Sobolev spaces to be Fredholm.","PeriodicalId":50104,"journal":{"name":"Journal of Operator Theory","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2019-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41426220","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}
Pub Date : 2019-11-01DOI: 10.7900/jot.2019oct09.2281
Octavio Arizmendi, J. Mingo
We show that using the cyclic group the transpose of an R-cyclic matrix can be decomposed along diagonal parts into a sum of parts which are freely independent over diagonal scalar matrices. Moreover, if the R-cyclic matrix is self-adjoint then the off-diagonal parts are R-diagonal.
{"title":"The cyclic group and the transpose of an R-cyclic matrix","authors":"Octavio Arizmendi, J. Mingo","doi":"10.7900/jot.2019oct09.2281","DOIUrl":"https://doi.org/10.7900/jot.2019oct09.2281","url":null,"abstract":"We show that using the cyclic group the transpose of an R-cyclic matrix can be decomposed along diagonal parts into a sum of parts which are freely independent over diagonal scalar matrices. Moreover, if the R-cyclic matrix is self-adjoint then the off-diagonal parts are R-diagonal.","PeriodicalId":50104,"journal":{"name":"Journal of Operator Theory","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2019-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44830994","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}
Pub Date : 2019-10-31DOI: 10.7900/jot.2019oct30.2265
A. Connes, C. Consani
We reconcile, at the semi-classical level, the original spectral realization of zeros of the Riemann zeta function as an ``absorption'' picture using the ad`ele class space, with the ``emission'' semi-classical computations of Berry and Keating. We then use the quantized calculus to analyse the recent attempt of X.-J.~Li at proving Weil's positivity, and explain its limit. Finally, we propose an operator theoretic semi-local framework directly related to the Riemann hypothesis.
{"title":"The scaling Hamiltonian","authors":"A. Connes, C. Consani","doi":"10.7900/jot.2019oct30.2265","DOIUrl":"https://doi.org/10.7900/jot.2019oct30.2265","url":null,"abstract":"We reconcile, at the semi-classical level, the original spectral realization of zeros of the Riemann zeta function as an ``absorption'' picture using the ad`ele class space, with the ``emission'' semi-classical computations of Berry and Keating. We then use the quantized calculus to analyse the recent attempt of X.-J.~Li at proving Weil's positivity, and explain its limit. Finally, we propose an operator theoretic semi-local framework directly related to the Riemann hypothesis.","PeriodicalId":50104,"journal":{"name":"Journal of Operator Theory","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2019-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48903688","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}
Pub Date : 2019-10-31DOI: 10.7900/jot.2019nov01.2279
S. Popa
A II1 factor M has the textit{stable single generation} (textit{SSG}) property if any amplification Mt, t>0, can be generated as a von Neumann algebra by a single element. We discuss a conjecture stating that if M is SSG, then M has a textit{tight} decomposition, i.e., there exists a pair of hyperfinite II1 subfactors R0,R1⊂M such that R0∨Rop1=B(L2M). We provide supporting evidence, explain why the conjecture is interesting, and discuss possible approaches to settle it. We also prove some related results.
{"title":"Tight decomposition of factors and the single generation problem","authors":"S. Popa","doi":"10.7900/jot.2019nov01.2279","DOIUrl":"https://doi.org/10.7900/jot.2019nov01.2279","url":null,"abstract":"A II1 factor M has the textit{stable single generation} (textit{SSG}) property if any amplification Mt, t>0, can be generated as a von Neumann algebra by a single element. We discuss a conjecture stating that if M is SSG, then M has a textit{tight} decomposition, i.e., there exists a pair of hyperfinite II1 subfactors R0,R1⊂M such that R0∨Rop1=B(L2M). We provide supporting evidence, explain why the conjecture is interesting, and discuss possible approaches to settle it. We also prove some related results.","PeriodicalId":50104,"journal":{"name":"Journal of Operator Theory","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2019-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48690683","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}
Pub Date : 2019-10-29DOI: 10.7900/jot.2020oct26.2293
D. Ursu
The notion of a plump subgroup was recently introduced by Amrutam. This is a relativized version of Powers' averaging property, and it is known that Powers' averaging property is equivalent to C∗-simplicity. With this in mind, we introduce a relativized notion of C∗-simplicity, and show that for normal subgroups it is equivalent to plumpness, along with several other characterizations.
{"title":"Relative C∗-simplicity and characterizations for normal subgroups","authors":"D. Ursu","doi":"10.7900/jot.2020oct26.2293","DOIUrl":"https://doi.org/10.7900/jot.2020oct26.2293","url":null,"abstract":"The notion of a plump subgroup was recently introduced by Amrutam. This is a relativized version of Powers' averaging property, and it is known that Powers' averaging property is equivalent to C∗-simplicity. With this in mind, we introduce a relativized notion of C∗-simplicity, and show that for normal subgroups it is equivalent to plumpness, along with several other characterizations.","PeriodicalId":50104,"journal":{"name":"Journal of Operator Theory","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2019-10-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48394200","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}
Pub Date : 2019-10-16DOI: 10.7900/jot.2019oct11.2281
Tobias Mai, R. Speicher
In 2000, Voiculescu proved an algebraic characterization of cyclic gradients of noncommutative polynomials. We extend this remarkable result in two different directions: first, we obtain an analogous characterization of free gradients; second, we lift both of these results to Voiculescu's fundamental framework of multivariable generalized difference quotient rings. For that purpose, we develop the concept of divergence operators, for both free and cyclic gradients, and study the associated (weak) grading and cyclic symmetrization operators, respectively. On the one hand, this puts a new complexion on the initial polynomial case, and on the other hand, it provides a uniform framework within which also other examples, such as a discrete version of the It^o stochastic integral, can be treated.
{"title":"A note on the free and cyclic differential calculus","authors":"Tobias Mai, R. Speicher","doi":"10.7900/jot.2019oct11.2281","DOIUrl":"https://doi.org/10.7900/jot.2019oct11.2281","url":null,"abstract":"In 2000, Voiculescu proved an algebraic characterization of cyclic gradients of noncommutative polynomials. We extend this remarkable result in two different directions: first, we obtain an analogous characterization of free gradients; second, we lift both of these results to Voiculescu's fundamental framework of multivariable generalized difference quotient rings. For that purpose, we develop the concept of divergence operators, for both free and cyclic gradients, and study the associated (weak) grading and cyclic symmetrization operators, respectively. On the one hand, this puts a new complexion on the initial polynomial case, and on the other hand, it provides a uniform framework within which also other examples, such as a discrete version of the It^o stochastic integral, can be treated.","PeriodicalId":50104,"journal":{"name":"Journal of Operator Theory","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2019-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45335365","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}
Pub Date : 2019-10-03DOI: 10.7900/jot.2020feb28.2292
I. Chalendar, J. Partington
This paper studies the behaviour of iterates of weigh-ted composition operators acting on spaces of analytic functions, with particular emphasis on the Hardy space H2. Questions relating to uniform, strong and weak convergence are resolved in many cases. Connected to this is the question when a weighted composition operator is an isometry, and new results are given in the case of the Hardy and Bergman spaces.
{"title":"Weighted composition operators: isometries and asymptotic behaviour","authors":"I. Chalendar, J. Partington","doi":"10.7900/jot.2020feb28.2292","DOIUrl":"https://doi.org/10.7900/jot.2020feb28.2292","url":null,"abstract":"This paper studies the behaviour of iterates of weigh-ted composition operators acting on spaces of analytic functions, with particular emphasis on the Hardy space H2. Questions relating to uniform, strong and weak convergence are resolved in many cases. Connected to this is the question when a weighted composition operator is an isometry, and new results are given in the case of the Hardy and Bergman spaces.","PeriodicalId":50104,"journal":{"name":"Journal of Operator Theory","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2019-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43356716","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}
Pub Date : 2019-09-15DOI: 10.7900/jot.2018may24.2226
Marcel Roman, A. Sandovici
An elementary construction of the square root of nonnegative selfadjoint linear relations in Hilbert spaces is presented.
给出了Hilbert空间中非负自伴线性关系的平方根的一个初等构造。
{"title":"The square root of nonnegative selfadjoint linear relations in Hilbert spaces","authors":"Marcel Roman, A. Sandovici","doi":"10.7900/jot.2018may24.2226","DOIUrl":"https://doi.org/10.7900/jot.2018may24.2226","url":null,"abstract":"An elementary construction of the square root of nonnegative selfadjoint linear relations in Hilbert spaces is presented.","PeriodicalId":50104,"journal":{"name":"Journal of Operator Theory","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2019-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46060311","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}
Pub Date : 2019-09-15DOI: 10.7900/jot.2018jun13.2222
Konrad Aguilar
We introduce a topology on the ideal space of any C∗-inductive limit built by an inverse limit of topologies and produce conditions for when this topology agrees with the Fell topology. With this topology, we impart criteria for when convergence of ideals of an AF-algebra can provide convergence of quotients in the quantum Gromov--Hausdorff propinquity building from previous joint work with Latr'{e}moli`{e}re. This bestows a continuous map from a class of ideals of the Boca--Mundici AF-algebra equipped with various topologies, including Jacobson and Fell topologies, to the space of quotients equipped with the propinquity topology.
{"title":"Fell topologies for AF-algebras and the quantum propinquity","authors":"Konrad Aguilar","doi":"10.7900/jot.2018jun13.2222","DOIUrl":"https://doi.org/10.7900/jot.2018jun13.2222","url":null,"abstract":"We introduce a topology on the ideal space of any C∗-inductive limit built by an inverse limit of topologies and produce conditions for when this topology agrees with the Fell topology. With this topology, we impart criteria for when convergence of ideals of an AF-algebra can provide convergence of quotients in the quantum Gromov--Hausdorff propinquity building from previous joint work with Latr'{e}moli`{e}re. This bestows a continuous map from a class of ideals of the Boca--Mundici AF-algebra equipped with various topologies, including Jacobson and Fell topologies, to the space of quotients equipped with the propinquity topology.","PeriodicalId":50104,"journal":{"name":"Journal of Operator Theory","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2019-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49603887","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}
Pub Date : 2019-09-07DOI: 10.7900/jot.2020oct07.2311
Tirthankar Bhattacharyya, H. Sau
We characterize the interpolating sequences and prove a Toeplitz--Corona theorem in the setting of bounded holomorphic functions on the symmetrized bidisk.
{"title":"Interpolating sequences and the Toeplitz--Corona theorem on the symmetrized bidisk","authors":"Tirthankar Bhattacharyya, H. Sau","doi":"10.7900/jot.2020oct07.2311","DOIUrl":"https://doi.org/10.7900/jot.2020oct07.2311","url":null,"abstract":"We characterize the interpolating sequences and prove a Toeplitz--Corona theorem in the setting of bounded holomorphic functions on the symmetrized bidisk.","PeriodicalId":50104,"journal":{"name":"Journal of Operator Theory","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2019-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43805380","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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