Pub Date : 2019-06-27DOI: 10.3318/pria.2019.119.05
J. Agler, John E. McCarthy, N. Young
A calcular algebra is a subalgebra of $H^infty(Omega)$ with norm given by $| phi | = sup | phi(T) |$ as $T$ ranges over a given class of commutative $d$-tuples of operators with Taylor spectrum in $O$. We discuss what algebras arise this way, and how they can be represented.
{"title":"Calcular algebras","authors":"J. Agler, John E. McCarthy, N. Young","doi":"10.3318/pria.2019.119.05","DOIUrl":"https://doi.org/10.3318/pria.2019.119.05","url":null,"abstract":"A calcular algebra is a subalgebra of $H^infty(Omega)$ with norm given by $| phi | = sup | phi(T) |$ as $T$ ranges over a given class of commutative $d$-tuples of operators with Taylor spectrum in $O$. We discuss what algebras arise this way, and how they can be represented.","PeriodicalId":434988,"journal":{"name":"Mathematical Proceedings of the Royal Irish Academy","volume":"112 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115739131","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}
Pub Date : 2018-10-25DOI: 10.3318/pria.2019.119.01
K. Hasegawa, Y. Ueda
This short note aims to give an insight to Arveson's boundary theorem by means of non-commutative Poisson boundaries and its applications.
这篇短文的目的是通过非交换泊松边界及其应用来理解Arveson边界定理。
{"title":"On Arveson's boundary theorem","authors":"K. Hasegawa, Y. Ueda","doi":"10.3318/pria.2019.119.01","DOIUrl":"https://doi.org/10.3318/pria.2019.119.01","url":null,"abstract":"This short note aims to give an insight to Arveson's boundary theorem by means of non-commutative Poisson boundaries and its applications.","PeriodicalId":434988,"journal":{"name":"Mathematical Proceedings of the Royal Irish Academy","volume":"4 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-10-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125596412","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}
Pub Date : 2018-05-06DOI: 10.3318/PRIA.2018.118.04
N. Hitchin
We interpret an open orbit in a 32-dimensional representation space of Spin(9,1) x SL(2,R) as a substitute for the non-existent group of invertible 2x2 matrices over the octonions and study various natural homogeneous subspaces. The approach is via twistor geometry in eight dimensions.
{"title":"SL(2) over the octonions","authors":"N. Hitchin","doi":"10.3318/PRIA.2018.118.04","DOIUrl":"https://doi.org/10.3318/PRIA.2018.118.04","url":null,"abstract":"We interpret an open orbit in a 32-dimensional representation space of Spin(9,1) x SL(2,R) as a substitute for the non-existent group of invertible 2x2 matrices over the octonions and study various natural homogeneous subspaces. The approach is via twistor geometry in eight dimensions.","PeriodicalId":434988,"journal":{"name":"Mathematical Proceedings of the Royal Irish Academy","volume":"96 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133496325","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}
Pub Date : 2018-04-08DOI: 10.3318/pria.2020.120.02
C. Kubrusly
The sets of strongly supercyclic, weakly l-sequentially supercyclic, weakly sequentially supercyclic, and weakly supercyclic vectors for an arbitrary normed-space operator are all dense in the normed space, regardless the notion of denseness one is considering, provided they are nonempty.
{"title":"Denseness of sets of supercyclic vectors","authors":"C. Kubrusly","doi":"10.3318/pria.2020.120.02","DOIUrl":"https://doi.org/10.3318/pria.2020.120.02","url":null,"abstract":"The sets of strongly supercyclic, weakly l-sequentially supercyclic, weakly sequentially supercyclic, and weakly supercyclic vectors for an arbitrary normed-space operator are all dense in the normed space, regardless the notion of denseness one is considering, provided they are nonempty.","PeriodicalId":434988,"journal":{"name":"Mathematical Proceedings of the Royal Irish Academy","volume":"6 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117083890","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}
Pub Date : 2018-01-24DOI: 10.3318/PRIA.2018.118.06
C. Kubrusly, B. Duggal
This paper provides conditions (i) to distinguish weak supercyclicity form supercyclicity for operators acting on normed and Banach spaces, and also (ii) to ensure when weak supercyclicity implies weak stability.
{"title":"On Weak Supercyclicity I","authors":"C. Kubrusly, B. Duggal","doi":"10.3318/PRIA.2018.118.06","DOIUrl":"https://doi.org/10.3318/PRIA.2018.118.06","url":null,"abstract":"This paper provides conditions (i) to distinguish weak supercyclicity form supercyclicity for operators acting on normed and Banach spaces, and also (ii) to ensure when weak supercyclicity implies weak stability.","PeriodicalId":434988,"journal":{"name":"Mathematical Proceedings of the Royal Irish Academy","volume":"3 2","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132791730","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}
Pub Date : 2017-10-07DOI: 10.3318/pria.2019.119.02
S. Basu, A. Singh
The notion of linear Hahn-Banach extension operator was first studied in detail by Heinrich and Mankiewicz (1982). Previously, J. Lindenstrauss (1966) studied similar versions of this notion in the context of non separable reflexive Banach spaces. Subsequently, Sims and Yost (1989) proved the existence of linear Hahn-Banach extension operators via interspersing subspaces in a purely Banach space theoretic set up. In this paper, we study similar questions in the context of Banach modules and module homomorphisms, in particular, Banach algebras of operators on Banach spaces. Based on Dales, Kania, Kochanek, Kozmider and Laustsen(2013), and also Kania and Laustsen (2017), we give complete answers for reflexive Banach spaces and the non-reflexive space constructed by Kania and Laustsen from the celebrated Argyros-Haydon's space with few operators.
{"title":"Linear Hahn-Banach type extension operators in Banach algebras of operators","authors":"S. Basu, A. Singh","doi":"10.3318/pria.2019.119.02","DOIUrl":"https://doi.org/10.3318/pria.2019.119.02","url":null,"abstract":"The notion of linear Hahn-Banach extension operator was first studied in detail by Heinrich and Mankiewicz (1982). Previously, J. Lindenstrauss (1966) studied similar versions of this notion in the context of non separable reflexive Banach spaces. Subsequently, Sims and Yost (1989) proved the existence of linear Hahn-Banach extension operators via interspersing subspaces in a purely Banach space theoretic set up. In this paper, we study similar questions in the context of Banach modules and module homomorphisms, in particular, Banach algebras of operators on Banach spaces. Based on Dales, Kania, Kochanek, Kozmider and Laustsen(2013), and also Kania and Laustsen (2017), we give complete answers for reflexive Banach spaces and the non-reflexive space constructed by Kania and Laustsen from the celebrated Argyros-Haydon's space with few operators.","PeriodicalId":434988,"journal":{"name":"Mathematical Proceedings of the Royal Irish Academy","volume":"24 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-10-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122925479","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}
Pub Date : 2017-03-01DOI: 10.3318/PRIA.2017.117.06
P. Hintz, M. Zworski
The purpose of this expository note is to revisit Morawetz's method for obtaining a lower bound on the rate of exponential decay of waves for the Dirichlet problem outside star-shaped obstacles, and to discuss the uniqueness of the sphere as the extremizer of the sharp lower bound proved by Ralston.
{"title":"Wave decay for star-shaped obstacles in ℝ3: papers of Morawetz and Ralston revisited","authors":"P. Hintz, M. Zworski","doi":"10.3318/PRIA.2017.117.06","DOIUrl":"https://doi.org/10.3318/PRIA.2017.117.06","url":null,"abstract":"The purpose of this expository note is to revisit Morawetz's method for obtaining a lower bound on the rate of exponential decay of waves for the Dirichlet problem outside star-shaped obstacles, and to discuss the uniqueness of the sphere as the extremizer of the sharp lower bound proved by Ralston.","PeriodicalId":434988,"journal":{"name":"Mathematical Proceedings of the Royal Irish Academy","volume":"35 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121098838","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}
Pub Date : 2016-05-17DOI: 10.3318/PRIA.2018.118.01
G. Larotonda
Let $p>1$ and $1/p+1/q=1$. Consider H"older's inequality $$ |ab^*|_1le |a|_p|b|_q $$ for the $p$-norms of some trace ($a,b$ are matrices, compact operators, elements of a finite $C^*$-algebra or a semi-finite von Neumann algebra). This note contains a simple proof (based on the case $p=2$) of the fact that equality holds iff $|a|^p=lambda |b|^q$ for some $lambdage 0$.
{"title":"The case of equality in Hölder's inequality for matrices and operators","authors":"G. Larotonda","doi":"10.3318/PRIA.2018.118.01","DOIUrl":"https://doi.org/10.3318/PRIA.2018.118.01","url":null,"abstract":"Let $p>1$ and $1/p+1/q=1$. Consider H\"older's inequality $$ |ab^*|_1le |a|_p|b|_q $$ for the $p$-norms of some trace ($a,b$ are matrices, compact operators, elements of a finite $C^*$-algebra or a semi-finite von Neumann algebra). This note contains a simple proof (based on the case $p=2$) of the fact that equality holds iff $|a|^p=lambda |b|^q$ for some $lambdage 0$.","PeriodicalId":434988,"journal":{"name":"Mathematical Proceedings of the Royal Irish Academy","volume":"32 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-05-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124907533","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}
Pub Date : 2016-03-22DOI: 10.3318/pria.2016.116.08
L. G. Brown
We show that ||u*u - v*v|| leq ||u - v|| for partial isometries u and v. There is a stronger inequality if both u and v are extreme points of the unit ball of a C*-algebra, and both inequalities are sharp. If u and v are partial isometries in a C*-algebra A such that ||u - v|| < 1, then u and v are homotopic through partial isometries in A. If both u and v are extremal, then it is sufficient that ||u - v|| < 2. The constants 1 and 2 are both sharp. We also discuss the continuity points of the map which assigns to each closed range element of A the partial isometry in its canonical polar decomposition.
{"title":"Some Metric and Homotopy Properties of Partial Isometries","authors":"L. G. Brown","doi":"10.3318/pria.2016.116.08","DOIUrl":"https://doi.org/10.3318/pria.2016.116.08","url":null,"abstract":"We show that ||u*u - v*v|| leq ||u - v|| for partial isometries u and v. There is a stronger inequality if both u and v are extreme points of the unit ball of a C*-algebra, and both inequalities are sharp. If u and v are partial isometries in a C*-algebra A such that ||u - v|| < 1, then u and v are homotopic through partial isometries in A. If both u and v are extremal, then it is sufficient that ||u - v|| < 2. The constants 1 and 2 are both sharp. We also discuss the continuity points of the map which assigns to each closed range element of A the partial isometry in its canonical polar decomposition.","PeriodicalId":434988,"journal":{"name":"Mathematical Proceedings of the Royal Irish Academy","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-03-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122808423","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}
Pub Date : 2015-11-29DOI: 10.3318/PRIA.2017.117.04
S. Power
An elementary proof is given to show that a parametrised algebraic curve in the plane may be traced out, in the sense of A. B. Kempe, by a finite pinned linkage. Additionally it is shown that any parametrised continuous curve gamma: [0,1] to R^2 may be traced out by an infinite linkage where the valencies of the joints is uniformly bounded. We also discuss related Kempe universality theorems and give a novel correction of Kempe's original argument.
给出了一个初等证明，证明了在a . B. Kempe意义下，平面上的参数化代数曲线可以用有限钉住连杆来描出。此外，还证明了任意参数化连续曲线gamma:[0,1]到R^2都可以被一个无限连杆跟踪，其中关节的价是一致有界的。我们还讨论了相关的肯普普适性定理，并对肯普的原论点进行了新的修正。
{"title":"Elementary proofs of Kempe universality","authors":"S. Power","doi":"10.3318/PRIA.2017.117.04","DOIUrl":"https://doi.org/10.3318/PRIA.2017.117.04","url":null,"abstract":"An elementary proof is given to show that a parametrised algebraic curve in the plane may be traced out, in the sense of A. B. Kempe, by a finite pinned linkage. Additionally it is shown that any parametrised continuous curve gamma: [0,1] to R^2 may be traced out by an infinite linkage where the valencies of the joints is uniformly bounded. We also discuss related Kempe universality theorems and give a novel correction of Kempe's original argument.","PeriodicalId":434988,"journal":{"name":"Mathematical Proceedings of the Royal Irish Academy","volume":"22 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2015-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126332777","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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