Mathematical Proceedings of the Royal Irish Academy最新文献

{"title":"AN UNCERTAINTY PRINCIPLE RELATED TO THE EUCLIDEAN MOTION GROUP","authors":"J. Christensen, H. Schlichtkrull","doi":"10.3318/PRIA.2004.104.2.249","DOIUrl":"https://doi.org/10.3318/PRIA.2004.104.2.249","url":null,"abstract":"We show that a well-known uncertainty principle for functions on the circle can be derived from an uncertainty principle for the Euclidean motion group. 1. Uncertainty principles related to Lie group representations Let G be a Lie group with Lie algebra g, and let (n, H) be a unitary representation of G. Then each element Xe g generates a closed, skew-adjoint operator n(X) on H by n(exptX)x-x n(X)x = hm '-o t with domain D(n(X)) consisting of all xe H for which the limit exists. The uncertainty principle related to n says that for operators generated by X, Y and [X, Y] the following holds ''n(X)x''''n(Y)x''>^'(n([X,Y])x,x)' for all xeD(n(X))nD(n(Y))nD(n([X, Y])). We would like to advocate this as a natural way to achieve uncertainty principles. It was first proposed in Kraus [4] for Lie groups with three dimensions or fewer, and the dimension constraint was recently removed by Christensen in [2]. For example, the classical Heisenberg uncertainty principle for functions on Un is easily derived in this way from the Schrodinger representation of the Heisenberg group (see [3, p. 212]). It is the purpose of this note to point out that the uncertainty principle for the circle, which was motivated by Breitenberger [1] and further discussed in [5; 6; 7; 8; 9; 10] is obtained similarly from the principal series representation of the Euclidean motion group of U2. * Corresponding author; e-mail: vepjan@math.ku.dk Mathematical Proceedings of the Royal Irish Academy, 104A (2), 249-252 (2004) © Royal Irish Academy This content downloaded from 157.55.39.58 on Tue, 15 Nov 2016 04:05:14 UTC All use subject to http://about.jstor.org/terms 250 Mathematical Proceedings of the Royal Irish Academy 2. The Euclidean motion group and a unitary representation Let G be the Euclidean motion group G=j(r,z)=(^or ^|rER,zecj Its Lie algebra is fl={(j j)|reR,zec} Let H be the Hubert space H = L2(J) of square integrable functions on the circle T = {s e C| 's' = 1}, with inner product (/\", g) = JT f(t)g(t)dt. As in [1 1, chapter V] the following defines a unitary representation of G on ^i: 7Tfl(r, z)/(?) = ^Re(z^/(^-^), (j e T) where a e C. For simplicity we assume in the following that a = l, which is sufficient for our purpose. The representation n' will be denoted n. 3. Operators generated from the representation We now generate three operators from elements of the Lie algebra g. Let X,YU r2egbe *-G s), \".-(s i) ^(o ?) then exp(ijr)=^ J), exParl)=^ i) and QxV(tY2)=(^ ^ and we then get 7t(X)f(s) = lim i-+0 / i->0 ? where /'(a) = jtf{eus). This operator has domain {/eL2(T)|?h-»/V0 absolutely continuous with /7 e L2(T)} (3.1) Also rv^^ v hm 1*0, t)f(s) f(s) hm e^s)fis) fis) . ,_, . 7t( rv^^ Fj)/^) = v hm i-o t t-+o t and /VUM r hm 40, iY)/(j) /(j) hm r e^fis) fis) . . . _. _ . . ni /VUM Y2)fis) = hm r t^o i /->o t when s=e10 . Both of these operators are defined on H. This content downloaded from 157.55.39.58 on Tue, 15 Nov 2016 04:05:14 UTC All use subject to http://about.jstor.o","PeriodicalId":434988,"journal":{"name":"Mathematical Proceedings of the Royal Irish Academy","volume":"168 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2004-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123275204","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":"THE HOMOLOGY OF HEISENBERG LIE ALGEBRAS OVER FIELDS OF CHARACTERISTIC TWO","authors":"Emil Sköldberg","doi":"10.3318/PRIA.2005.105.2.47","DOIUrl":"https://doi.org/10.3318/PRIA.2005.105.2.47","url":null,"abstract":"The generating function of the Betti numbers of the Heisenberg Lie algebra over a field of characteristic 2 is calculated using discrete Morse theory.","PeriodicalId":434988,"journal":{"name":"Mathematical Proceedings of the Royal Irish Academy","volume":"23 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2003-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132723279","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":"COTYPE AND NONLINEAR ABSOLUTELY SUMMING MAPPINGS","authors":"D. Pellegrino","doi":"10.3318/PRIA.2005.105.1.75","DOIUrl":"https://doi.org/10.3318/PRIA.2005.105.1.75","url":null,"abstract":"In this paper we study absolutely summing mappings on Banach spaces by exploring the cotype of their domains and ranges. We give new examples of absolutely summing analytic mappings and polynomial/multilinear versions of linear coincidence theorems.","PeriodicalId":434988,"journal":{"name":"Mathematical Proceedings of the Royal Irish Academy","volume":"36 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2003-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127608224","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":"ON THE JACOBI EQUATION AND MANIFOLDS WITH MULTIPLE CONJUGATE POINTS","authors":"J. Burns, Eoghan J. Staunton, D. Wraith","doi":"10.3318/PRIA.2013.113.03","DOIUrl":"https://doi.org/10.3318/PRIA.2013.113.03","url":null,"abstract":"We investigate the phenomenon of multiple conjugate points along \u0000a geodesic. In the first instance, we investigate conjugate points in the context \u0000of the Jacobi equation, a second order ordinary differential equation, which captures precisely the geometry of conjugate points on surfaces. We then construct \u0000geometric examples which exhibit similar properties in higher dimensions.","PeriodicalId":434988,"journal":{"name":"Mathematical Proceedings of the Royal Irish Academy","volume":"6 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115708448","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":"Geometry of vector arrays and operator ampliations in Hilbert\u0000 spaces","authors":"Klímek","doi":"10.3318/pria.2019.119.09","DOIUrl":"https://doi.org/10.3318/pria.2019.119.09","url":null,"abstract":"","PeriodicalId":434988,"journal":{"name":"Mathematical Proceedings of the Royal Irish Academy","volume":"50 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116912745","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 CONTRIBUTION TO THE INVERSE EIGENVALUE PROBLEM FOR NON-NEGATIVE MATRICES","authors":"F. Holland","doi":"10.3318/PRIA.2013.113.09","DOIUrl":"https://doi.org/10.3318/PRIA.2013.113.09","url":null,"abstract":"Sometime in the early 1970s Trevor introduced me to the spectral theory of positive linear operators which owes its origins to the celebrated Perron-Frobenius theorem according to which the spectral radius of a non-negative matrix is one of its eigenvalues, and possesses a corresponding eigenvector whose components are nonnegative real numbers. This was a subject dear to his heart, and a recurring theme to which he often returned in later years. In this article we make a tangential contribution to the converse of the PerronFrobenius theorem, the so-called inverse eigenvalue problem for non-negative matrices, namely, under what circumstances are the components of a vector of complex numbers the eigenvalues of such a matrix? To this end, we associate with each vector of unit norm an analytic self map of the unit open disc of the complex plane, which is also a rational function, and develop its power series expansion about the origin. Sufficient conditions are presented that ensure that the resulting coefficients which encode information about the chosen vector are non-negative. Conversely, if these are all non-negative, it turns out that the vector satisfies conditions that are necessary ones for it to solve the inverse problem.","PeriodicalId":434988,"journal":{"name":"Mathematical Proceedings of the Royal Irish Academy","volume":"67 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125867185","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 GENERAL WOLFF THEOREM FOR ARBITRARY BANACH SPACES","authors":"P. Mellon","doi":"10.1353/mpr.2004.0009","DOIUrl":"https://doi.org/10.1353/mpr.2004.0009","url":null,"abstract":"","PeriodicalId":434988,"journal":{"name":"Mathematical Proceedings of the Royal Irish Academy","volume":"42 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126044609","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":"Dominating Sets for Analytic and Harmonic Functions and Completeness of Weighted Bergman Spaces","authors":"N. Arcozzi, Anders Björn","doi":"10.3318/PRIA.2002.102.2.175","DOIUrl":"https://doi.org/10.3318/PRIA.2002.102.2.175","url":null,"abstract":"A set E c Q is holomorphically dominating for Q if sUP,cE If(z)I = SUpzcn If(z)l for all holomorphic functions f on Q. As follows from a result of Stray, this property is equivalent to the inaccessibility of the Aleksandrov compactification point * (of Q) from Q x E. Moreover, it is equivalent to a large number of other statements (old and new) of holomorphic, harmonic and topological nature, including that a certain weighted Bergman space with p = oo is a Banach space. We extend this to the cases of harmonic functions in R' and holomorphic functions in cV. We also present some results on when weighted Bergman spaces are (quasi)-Banach spaces, the case p = oo being characterised by the result mentioned above.","PeriodicalId":434988,"journal":{"name":"Mathematical Proceedings of the Royal Irish Academy","volume":"49 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122710930","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":"EDITORIAL: AN HOMAGE TO TREVOR WEST","authors":"M. Mathieu","doi":"10.3318/pria.2013.113.06","DOIUrl":"https://doi.org/10.3318/pria.2013.113.06","url":null,"abstract":"","PeriodicalId":434988,"journal":{"name":"Mathematical Proceedings of the Royal Irish Academy","volume":"22 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122635725","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":"SOME VARIANTS OF WEBER'S THEOREM","authors":"S. Mécheri","doi":"10.3318/PRIA.2004.104.1.67","DOIUrl":"https://doi.org/10.3318/PRIA.2004.104.1.67","url":null,"abstract":"Weber’s theorem says that if A : H 0 H is bounded and linear on a separable Hilbert space, then any operator that is compact, commutes with A and lies in the weak closure of the range of the inner derivation induced by A must also be quasinilpotent. In this note we consider related problems for generalised inner derivations associated with operators A and B on H.","PeriodicalId":434988,"journal":{"name":"Mathematical Proceedings of the Royal Irish Academy","volume":"16 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114560273","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}