Abstract In this paper we apply the results obtained in [3] to establish some outcomes of the study of the behaviour of a class of linear operators, which include the Sylvester ones, acting on slice semi-regular functions. We first present a detailed study of the kernel of the linear operator ℒf,g (when not trivial), showing that it has dimension 2 if exactly one between f and g is a zero divisor, and it has dimension 3 if both f and g are zero divisors. Afterwards, we deepen the analysis of the behaviour of the -product, giving a complete classification of the cases when the functions fv, gv and fv gv are linearly dependent and obtaining, as a by-product, a necessary and sufficient condition on the functions f and g in order their *-product is slice-preserving. At last, we give an Embry-type result which classifies the functions f and g such that for any function h commuting with f + g and f * g, we have that h commutes with f and g, too.
{"title":"Applications of the Sylvester operator in the space of slice semi-regular functions","authors":"Altavilla Amedeo, C. de Fabritiis","doi":"10.1515/conop-2020-0001","DOIUrl":"https://doi.org/10.1515/conop-2020-0001","url":null,"abstract":"Abstract In this paper we apply the results obtained in [3] to establish some outcomes of the study of the behaviour of a class of linear operators, which include the Sylvester ones, acting on slice semi-regular functions. We first present a detailed study of the kernel of the linear operator ℒf,g (when not trivial), showing that it has dimension 2 if exactly one between f and g is a zero divisor, and it has dimension 3 if both f and g are zero divisors. Afterwards, we deepen the analysis of the behaviour of the -product, giving a complete classification of the cases when the functions fv, gv and fv gv are linearly dependent and obtaining, as a by-product, a necessary and sufficient condition on the functions f and g in order their *-product is slice-preserving. At last, we give an Embry-type result which classifies the functions f and g such that for any function h commuting with f + g and f * g, we have that h commutes with f and g, too.","PeriodicalId":53800,"journal":{"name":"Concrete Operators","volume":"7 1","pages":"1 - 12"},"PeriodicalIF":0.6,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/conop-2020-0001","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48746531","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}
Abstract This survey shows how, for the Nevanlinna class 𝒩 of the unit disc, one can define and often characterize the analogues of well-known objects and properties related to the algebra of bounded analytic functions ℋ∞: interpolating sequences, Corona theorem, sets of determination, stable rank, as well as the more recent notions of Weak Embedding Property and threshold of invertibility for quotient algebras. The general rule we observe is that a given result for ℋ∞ can be transposed to 𝒩 by replacing uniform bounds by a suitable control by positive harmonic functions. We show several instances where this rule applies, as well as some exceptions. We also briefly discuss the situation for the related Smirnov class.
{"title":"From ℋ∞ to 𝒩. Pointwise properties and algebraic structure in the Nevanlinna class","authors":"X. Massaneda, P. Thomas","doi":"10.1515/conop-2020-0007","DOIUrl":"https://doi.org/10.1515/conop-2020-0007","url":null,"abstract":"Abstract This survey shows how, for the Nevanlinna class 𝒩 of the unit disc, one can define and often characterize the analogues of well-known objects and properties related to the algebra of bounded analytic functions ℋ∞: interpolating sequences, Corona theorem, sets of determination, stable rank, as well as the more recent notions of Weak Embedding Property and threshold of invertibility for quotient algebras. The general rule we observe is that a given result for ℋ∞ can be transposed to 𝒩 by replacing uniform bounds by a suitable control by positive harmonic functions. We show several instances where this rule applies, as well as some exceptions. We also briefly discuss the situation for the related Smirnov class.","PeriodicalId":53800,"journal":{"name":"Concrete Operators","volume":"7 1","pages":"91 - 115"},"PeriodicalIF":0.6,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/conop-2020-0007","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66888745","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}
Abstract The author has computed the bounds of the Hilbert operator on some sequence spaces [18, 19]. Through this study the author has investigated the bounds of operators on the Hilbert sequence space and the present study is a complement of those previous research.
{"title":"Bounds of operators on the Hilbert sequence space","authors":"H. Roopaei","doi":"10.1515/conop-2020-0104","DOIUrl":"https://doi.org/10.1515/conop-2020-0104","url":null,"abstract":"Abstract The author has computed the bounds of the Hilbert operator on some sequence spaces [18, 19]. Through this study the author has investigated the bounds of operators on the Hilbert sequence space and the present study is a complement of those previous research.","PeriodicalId":53800,"journal":{"name":"Concrete Operators","volume":"91 6","pages":"155 - 165"},"PeriodicalIF":0.6,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/conop-2020-0104","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41259915","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}
Abstract We present a survey of some recent results concerning joint numerical ranges of n-tuples of Hilbert space operators, accompanied with several new observations and remarks. Thereafter, numerical ranges techniques will be applied to various problems of operator theory. In particular, we discuss problems concerning orbits of operators, diagonals of operators and their tuples, and pinching problems. Lastly, motivated by known results on the numerical radius of a single operator, we examine whether, given bounded linear operators T1, . . ., Tn on a Hilbert space H, there exists a unit vector x ∈ H such that |〈Tjx, x〉| is “large” for all j = 1, . . . , n.
摘要本文综述了最近关于Hilbert空间算子n元组联合数值范围的一些结果，并给出了一些新的观察和注释。此后，数值范围技术将应用于算子理论的各种问题。特别地，我们讨论了算子的轨道问题，算子的对角线及其元组问题，以及捏紧问题。最后，根据关于单个算子数值半径的已知结果，我们检验了在Hilbert空间H上，给定有界线性算子T1，…，Tn，是否存在一个单位向量x∈H，使得| < Tjx, x > |对于所有j = 1，…都是“大”的。, n。
{"title":"Joint numerical ranges: recent advances and applications minicourse by V. Müller and Yu. Tomilov","authors":"V. Müller, Y. Tomilov","doi":"10.1515/conop-2020-0102","DOIUrl":"https://doi.org/10.1515/conop-2020-0102","url":null,"abstract":"Abstract We present a survey of some recent results concerning joint numerical ranges of n-tuples of Hilbert space operators, accompanied with several new observations and remarks. Thereafter, numerical ranges techniques will be applied to various problems of operator theory. In particular, we discuss problems concerning orbits of operators, diagonals of operators and their tuples, and pinching problems. Lastly, motivated by known results on the numerical radius of a single operator, we examine whether, given bounded linear operators T1, . . ., Tn on a Hilbert space H, there exists a unit vector x ∈ H such that |〈Tjx, x〉| is “large” for all j = 1, . . . , n.","PeriodicalId":53800,"journal":{"name":"Concrete Operators","volume":"7 1","pages":"133 - 154"},"PeriodicalIF":0.6,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/conop-2020-0102","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48040870","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}
Abstract We give a survey on approximation numbers of composition operators on the Hardy space, on the disk and on the polydisk, and add corresponding new results on their entropy numbers, revealing how they are different.
{"title":"Approximation and entropy numbers of composition operators","authors":"Daniel Li, H. Queffélec, L. Rodríguez-Piazza","doi":"10.1515/conop-2020-0106","DOIUrl":"https://doi.org/10.1515/conop-2020-0106","url":null,"abstract":"Abstract We give a survey on approximation numbers of composition operators on the Hardy space, on the disk and on the polydisk, and add corresponding new results on their entropy numbers, revealing how they are different.","PeriodicalId":53800,"journal":{"name":"Concrete Operators","volume":"7 1","pages":"166 - 179"},"PeriodicalIF":0.6,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/conop-2020-0106","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43624388","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}
Abstract In this note, we study defect operators in the case of holomorphic functions of the unit ball of ℂn. These operators are built from weighted Bergman kernel with a holomorphic vector. We obtain a description of sub-Hilbert spaces and we give a sufficient condition so that theses spaces are the same.
{"title":"Weighted Sub-Bergman Hilbert spaces in the unit ball of ℂn","authors":"R. Rososzczuk, F. Symesak","doi":"10.1515/conop-2020-0103","DOIUrl":"https://doi.org/10.1515/conop-2020-0103","url":null,"abstract":"Abstract In this note, we study defect operators in the case of holomorphic functions of the unit ball of ℂn. These operators are built from weighted Bergman kernel with a holomorphic vector. We obtain a description of sub-Hilbert spaces and we give a sufficient condition so that theses spaces are the same.","PeriodicalId":53800,"journal":{"name":"Concrete Operators","volume":"7 1","pages":"124 - 132"},"PeriodicalIF":0.6,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/conop-2020-0103","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42424333","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}
Abstract Let 𝕋 be a homogeneous tree. We prove that if f ∈ Lp(𝕋), 1 ≤ p ≤ 2, then the Riesz means SzR (f) converge to f everywhere as R → ∞, whenever Re z > 0.
摘要:设其一为齐次树。证明了如果f∈Lp(f)， 1≤p≤2，则Riesz意味着SzR (f)处处收敛于f，当R→∞时，当Re z > 0时。
{"title":"Riesz means on homogeneous trees","authors":"E. Papageorgiou","doi":"10.1515/conop-2020-0111","DOIUrl":"https://doi.org/10.1515/conop-2020-0111","url":null,"abstract":"Abstract Let 𝕋 be a homogeneous tree. We prove that if f ∈ Lp(𝕋), 1 ≤ p ≤ 2, then the Riesz means SzR (f) converge to f everywhere as R → ∞, whenever Re z > 0.","PeriodicalId":53800,"journal":{"name":"Concrete Operators","volume":"8 1","pages":"60 - 65"},"PeriodicalIF":0.6,"publicationDate":"2019-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/conop-2020-0111","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42972388","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}
Abstract In this paper we study Hausdorff operators on the Bergman spaces Ap(𝕌) of the upper half plane.
本文研究了上半平面Bergman空间Ap()上的Hausdorff算子。
{"title":"Hausdorff operators on Bergman spaces of the upper half plane","authors":"G. Stylogiannis","doi":"10.1515/conop-2020-0005","DOIUrl":"https://doi.org/10.1515/conop-2020-0005","url":null,"abstract":"Abstract In this paper we study Hausdorff operators on the Bergman spaces Ap(𝕌) of the upper half plane.","PeriodicalId":53800,"journal":{"name":"Concrete Operators","volume":"7 1","pages":"69 - 80"},"PeriodicalIF":0.6,"publicationDate":"2019-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/conop-2020-0005","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43855177","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}
Abstract Let w be a semiclassical weight that is generic in Magnus’s sense, and (pn)n=0∞ ({p_n})_{n = 0}^infty the corresponding sequence of orthogonal polynomials. We express the Christoffel–Darboux kernel as a sum of products of Hankel integral operators. For ψ ∈ L∞ (iℝ), let W(ψ) be the Wiener-Hopf operator with symbol ψ. We give sufficient conditions on ψ such that 1/ det W(ψ) W(ψ−1) = det(I − Γϕ1 Γϕ2) where Γϕ1 and Γϕ2 are Hankel operators that are Hilbert–Schmidt. For certain, ψ Barnes’s integral leads to an expansion of this determinant in terms of the generalised hypergeometric 2mF2m-1. These results extend those of Basor and Chen [2], who obtained 4F3 likewise. We include examples where the Wiener–Hopf factors are found explicitly.
{"title":"On Determinant Expansions for Hankel Operators","authors":"G. Blower, Yang Chen","doi":"10.1515/conop-2020-0002","DOIUrl":"https://doi.org/10.1515/conop-2020-0002","url":null,"abstract":"Abstract Let w be a semiclassical weight that is generic in Magnus’s sense, and (pn)n=0∞ ({p_n})_{n = 0}^infty the corresponding sequence of orthogonal polynomials. We express the Christoffel–Darboux kernel as a sum of products of Hankel integral operators. For ψ ∈ L∞ (iℝ), let W(ψ) be the Wiener-Hopf operator with symbol ψ. We give sufficient conditions on ψ such that 1/ det W(ψ) W(ψ−1) = det(I − Γϕ1 Γϕ2) where Γϕ1 and Γϕ2 are Hankel operators that are Hilbert–Schmidt. For certain, ψ Barnes’s integral leads to an expansion of this determinant in terms of the generalised hypergeometric 2mF2m-1. These results extend those of Basor and Chen [2], who obtained 4F3 likewise. We include examples where the Wiener–Hopf factors are found explicitly.","PeriodicalId":53800,"journal":{"name":"Concrete Operators","volume":"7 1","pages":"13 - 44"},"PeriodicalIF":0.6,"publicationDate":"2019-01-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/conop-2020-0002","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45543622","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}
Abstract We characterize closed range composition operators on the Dirichlet space for a particular class of composition symbols. The characterization relies on a result about Fredholm Toeplitz operators with BMO1 symbols, and with Berezin transforms of vanishing oscillation.
{"title":"On the closed range problem for composition operators on the Dirichlet space","authors":"N. Zorboska","doi":"10.1515/conop-2019-0007","DOIUrl":"https://doi.org/10.1515/conop-2019-0007","url":null,"abstract":"Abstract We characterize closed range composition operators on the Dirichlet space for a particular class of composition symbols. The characterization relies on a result about Fredholm Toeplitz operators with BMO1 symbols, and with Berezin transforms of vanishing oscillation.","PeriodicalId":53800,"journal":{"name":"Concrete Operators","volume":"6 1","pages":"76 - 81"},"PeriodicalIF":0.6,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/conop-2019-0007","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41448201","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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