Abstract The Invariant Subspace Problem for Hilbert spaces is a long-standing question and the use of universal operators in the sense of Rota has been an important tool for studying such important problem. In this survey, we focus on Rota’s universal operators, pointing out their main properties and exhibiting some old and recent examples.
{"title":"An introduction to Rota’s universal operators: properties, old and new examples and future issues","authors":"C. Cowen, E. Gallardo-Gutiérrez","doi":"10.1515/conop-2016-0006","DOIUrl":"https://doi.org/10.1515/conop-2016-0006","url":null,"abstract":"Abstract The Invariant Subspace Problem for Hilbert spaces is a long-standing question and the use of universal operators in the sense of Rota has been an important tool for studying such important problem. In this survey, we focus on Rota’s universal operators, pointing out their main properties and exhibiting some old and recent examples.","PeriodicalId":53800,"journal":{"name":"Concrete Operators","volume":"9 1","pages":"43 - 51"},"PeriodicalIF":0.6,"publicationDate":"2016-01-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/conop-2016-0006","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66888099","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 review article we present the problem of studying Hardy spaces and the related Szeg˝o projection on worm domains. We review the importance of the Diederich–Fornæss worm domain as a smooth bounded pseudoconvex domain whose Bergman projection does not preserve Sobolev spaces of sufficiently high order and we highlight which difficulties arise in studying the same problem for the Szeg˝o projection. Finally, we announce and discuss the results we have obtained so far in the setting of non-smooth worm domains.
{"title":"On Hardy spaces on worm domains","authors":"A. Monguzzi","doi":"10.1515/conop-2016-0005","DOIUrl":"https://doi.org/10.1515/conop-2016-0005","url":null,"abstract":"Abstract In this review article we present the problem of studying Hardy spaces and the related Szeg˝o projection on worm domains. We review the importance of the Diederich–Fornæss worm domain as a smooth bounded pseudoconvex domain whose Bergman projection does not preserve Sobolev spaces of sufficiently high order and we highlight which difficulties arise in studying the same problem for the Szeg˝o projection. Finally, we announce and discuss the results we have obtained so far in the setting of non-smooth worm domains.","PeriodicalId":53800,"journal":{"name":"Concrete Operators","volume":"3 1","pages":"29 - 42"},"PeriodicalIF":0.6,"publicationDate":"2016-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/conop-2016-0005","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66887997","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 Meromorphic Inner Functions (MIFs) on the upper half plane play an important role in applications to spectral problems for differential operators. In this paper, we survey some recent results concerning function theoretic properties of MIFs and show their connections with spectral problems for the Schrödinger operator.
{"title":"Restricted interpolation by meromorphic inner functions","authors":"A. Poltoratski, Rishika Rupam","doi":"10.1515/conop-2016-0012","DOIUrl":"https://doi.org/10.1515/conop-2016-0012","url":null,"abstract":"Abstract Meromorphic Inner Functions (MIFs) on the upper half plane play an important role in applications to spectral problems for differential operators. In this paper, we survey some recent results concerning function theoretic properties of MIFs and show their connections with spectral problems for the Schrödinger operator.","PeriodicalId":53800,"journal":{"name":"Concrete Operators","volume":"23 1","pages":"102 - 111"},"PeriodicalIF":0.6,"publicationDate":"2016-01-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/conop-2016-0012","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66888525","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 extend the Fuglede-Putnam theorem modulo the Hilbert-Schmidt class to almost normal operators with finite Hilbert-Schmidt modulus of quasi-triangularity.
{"title":"A Fuglede-Putnam theorem modulo the Hilbert-Schmidt class for almost normal operators with finite modulus of Hilbert-Schmidt quasi-triangularity","authors":"Vasile Lauric","doi":"10.1515/conop-2016-0002","DOIUrl":"https://doi.org/10.1515/conop-2016-0002","url":null,"abstract":"Abstract We extend the Fuglede-Putnam theorem modulo the Hilbert-Schmidt class to almost normal operators with finite Hilbert-Schmidt modulus of quasi-triangularity.","PeriodicalId":53800,"journal":{"name":"Concrete Operators","volume":"3 1","pages":"14 - 8"},"PeriodicalIF":0.6,"publicationDate":"2016-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/conop-2016-0002","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66887790","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}
N. Arcozzi, P. Mozolyako, Karl-Mikael Perfekt, S. Richter, G. Sarfatti
Abstract We study the reproducing kernel Hilbert space with kernel kd , where d is a positive integer and k is the reproducing kernel of the analytic Dirichlet space.
{"title":"Some Hilbert spaces related with the Dirichlet space","authors":"N. Arcozzi, P. Mozolyako, Karl-Mikael Perfekt, S. Richter, G. Sarfatti","doi":"10.1515/conop-2016-0011","DOIUrl":"https://doi.org/10.1515/conop-2016-0011","url":null,"abstract":"Abstract We study the reproducing kernel Hilbert space with kernel kd , where d is a positive integer and k is the reproducing kernel of the analytic Dirichlet space.","PeriodicalId":53800,"journal":{"name":"Concrete Operators","volume":"3 1","pages":"101 - 94"},"PeriodicalIF":0.6,"publicationDate":"2015-12-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/conop-2016-0011","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66888438","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 spaces of holomorphic functions on the right half-plane R, that we denote by Mpω, whose growth conditions are given in terms of a translation invariant measure ω on the closed half-plane R. Such a measure has the form ω = ν ⊗ m, where m is the Lebesgue measure on R and ν is a regular Borel measure on [0, +∞). We call these spaces generalized Hardy–Bergman spaces on the half-plane R. We study in particular the case of ν purely atomic, with point masses on an arithmetic progression on [0, +∞). We obtain a Paley–Wiener theorem for M2ω, and consequentely the expression for its reproducing kernel. We study the growth of functions in such space and in particular show that Mpω contains functions of order 1. Moreover, we prove that the orthogonal projection from Lp(R,dω) into Mpω is unbounded for p ≠ 2. Furthermore, we compare the spaces Mpω with the classical Hardy and Bergman spaces, and some other Hardy– Bergman-type spaces introduced more recently.
{"title":"On some spaces of holomorphic functions of exponential growth on a half-plane","authors":"M. Peloso, M. Salvatori","doi":"10.1515/conop-2016-0008","DOIUrl":"https://doi.org/10.1515/conop-2016-0008","url":null,"abstract":"Abstract In this paper we study spaces of holomorphic functions on the right half-plane R, that we denote by Mpω, whose growth conditions are given in terms of a translation invariant measure ω on the closed half-plane R. Such a measure has the form ω = ν ⊗ m, where m is the Lebesgue measure on R and ν is a regular Borel measure on [0, +∞). We call these spaces generalized Hardy–Bergman spaces on the half-plane R. We study in particular the case of ν purely atomic, with point masses on an arithmetic progression on [0, +∞). We obtain a Paley–Wiener theorem for M2ω, and consequentely the expression for its reproducing kernel. We study the growth of functions in such space and in particular show that Mpω contains functions of order 1. Moreover, we prove that the orthogonal projection from Lp(R,dω) into Mpω is unbounded for p ≠ 2. Furthermore, we compare the spaces Mpω with the classical Hardy and Bergman spaces, and some other Hardy– Bergman-type spaces introduced more recently.","PeriodicalId":53800,"journal":{"name":"Concrete Operators","volume":"3 1","pages":"52 - 67"},"PeriodicalIF":0.6,"publicationDate":"2015-12-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/conop-2016-0008","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66888210","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 A space of entire functions of several complex variables rapidly decreasing on Rn and such that their growth along iRn is majorized with the help of a family of weight functions is considered in this paper. For such space an equivalent description in terms of estimates on all of its partial derivatives as functions on Rn and a Paley-Wiener type theorem are obtained.
{"title":"On a space of entire functions rapidly decreasing on Rn and its Fourier transform","authors":"I. Musin","doi":"10.1515/conop-2015-0007","DOIUrl":"https://doi.org/10.1515/conop-2015-0007","url":null,"abstract":"Abstract A space of entire functions of several complex variables rapidly decreasing on Rn and such that their growth along iRn is majorized with the help of a family of weight functions is considered in this paper. For such space an equivalent description in terms of estimates on all of its partial derivatives as functions on Rn and a Paley-Wiener type theorem are obtained.","PeriodicalId":53800,"journal":{"name":"Concrete Operators","volume":"2 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2015-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/conop-2015-0007","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66887620","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 discuss non commutative functions, which naturally arise when dealing with functions of more than one matrix variable.
摘要讨论了在处理多个矩阵变量的函数时自然出现的非交换函数。
{"title":"Aspects of non-commutative function theory","authors":"J. Agler, John E. McCarthy","doi":"10.1515/conop-2016-0003","DOIUrl":"https://doi.org/10.1515/conop-2016-0003","url":null,"abstract":"Abstract We discuss non commutative functions, which naturally arise when dealing with functions of more than one matrix variable.","PeriodicalId":53800,"journal":{"name":"Concrete Operators","volume":"3 1","pages":"15 - 24"},"PeriodicalIF":0.6,"publicationDate":"2015-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/conop-2016-0003","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66887856","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 paperwe study both the spectra and the essential spectra ofweighted composition operators on Hardy spaces Hp(ⅅ), standard weighted Bergman spaces Apα(ⅅ) and weighted H∞1-type spaces when the symbols are of hyperbolic type
{"title":"Essential spectra of weighted composition operators with hyperbolic symbols","authors":"Olli Hyvärinen, Ilmari Nieminen","doi":"10.1515/conop-2015-0006","DOIUrl":"https://doi.org/10.1515/conop-2015-0006","url":null,"abstract":"Abstract In this paperwe study both the spectra and the essential spectra ofweighted composition operators on Hardy spaces Hp(ⅅ), standard weighted Bergman spaces Apα(ⅅ) and weighted H∞1-type spaces when the symbols are of hyperbolic type","PeriodicalId":53800,"journal":{"name":"Concrete Operators","volume":"2 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2015-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/conop-2015-0006","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66887535","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 investigate weighted inequalities for fractional maximal operators and fractional integral operators.We work within the innovative framework of “entropy bounds” introduced by Treil–Volberg. Using techniques developed by Lacey and the second author, we are able to efficiently prove the weighted inequalities.
{"title":"Entropy bump conditions for fractional maximal and integral operators","authors":"R. Rahm, Scott Spencer","doi":"10.1515/conop-2016-0013","DOIUrl":"https://doi.org/10.1515/conop-2016-0013","url":null,"abstract":"Abstract We investigate weighted inequalities for fractional maximal operators and fractional integral operators.We work within the innovative framework of “entropy bounds” introduced by Treil–Volberg. Using techniques developed by Lacey and the second author, we are able to efficiently prove the weighted inequalities.","PeriodicalId":53800,"journal":{"name":"Concrete Operators","volume":"3 1","pages":"112 - 121"},"PeriodicalIF":0.6,"publicationDate":"2015-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/conop-2016-0013","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66888607","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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