Abstract We study twoweight inequalities in the recent innovative language of ‘entropy’ due to Treil-Volberg. The inequalities are extended to Lp, for 1 < p ≠ 2 < ∞, with new short proofs. A result proved is as follows. Let ɛ be a monotonic increasing function on (1,∞) which satisfy Let σ and w be two weights on ℝd. If this supremum is finite, for a choice of 1 < p < ∞, then any Calderón-Zygmund operator T satisfies the bound ||Tof||Lp(w) ≲ ||f|| Lp(o).
摘要:我们研究了Treil-Volberg最近提出的“熵”这一创新语言中的两个权不等式。对1 < p≠2 <∞的不等式推广到Lp，并给出了新的简短证明。证明结果如下:设(1，∞)上的一个单调递增函数，满足σ和w是两个权值。如果这个上极值是有限的，对于1 < p <∞的选择，则任意Calderón-Zygmund算子T满足||Tof||Lp(w) > ||f|| Lp(o)。
{"title":"On Entropy Bumps for Calderón-Zygmund Operators","authors":"M. Lacey, Scott Spencer","doi":"10.1515/conop-2015-0003","DOIUrl":"https://doi.org/10.1515/conop-2015-0003","url":null,"abstract":"Abstract We study twoweight inequalities in the recent innovative language of ‘entropy’ due to Treil-Volberg. The inequalities are extended to Lp, for 1 < p ≠ 2 < ∞, with new short proofs. A result proved is as follows. Let ɛ be a monotonic increasing function on (1,∞) which satisfy Let σ and w be two weights on ℝd. If this supremum is finite, for a choice of 1 < p < ∞, then any Calderón-Zygmund operator T satisfies the bound ||Tof||Lp(w) ≲ ||f|| Lp(o).","PeriodicalId":53800,"journal":{"name":"Concrete Operators","volume":"2 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2015-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/conop-2015-0003","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66887105","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 MX,w(ℝ) denote the algebra of the Fourier multipliers on a separable weighted Banach function space X(ℝ,w).We prove that if the Cauchy singular integral operator S is bounded on X(ℝ, w), thenMX,w(ℝ) is continuously embedded into L∞(ℝ). An important consequence of the continuous embedding MX,w(ℝ) ⊂ L∞(ℝ) is that MX,w(ℝ) is a Banach algebra.
{"title":"Banach algebra of the Fourier multipliers on weighted Banach function spaces","authors":"A. Karlovich","doi":"10.1515/conop-2015-0001","DOIUrl":"https://doi.org/10.1515/conop-2015-0001","url":null,"abstract":"Abstract Let MX,w(ℝ) denote the algebra of the Fourier multipliers on a separable weighted Banach function space X(ℝ,w).We prove that if the Cauchy singular integral operator S is bounded on X(ℝ, w), thenMX,w(ℝ) is continuously embedded into L∞(ℝ). An important consequence of the continuous embedding MX,w(ℝ) ⊂ L∞(ℝ) is that MX,w(ℝ) is a Banach algebra.","PeriodicalId":53800,"journal":{"name":"Concrete Operators","volume":"2 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2015-03-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/conop-2015-0001","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66887434","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}
Daniel Li, Herv'e Queff'elec, Luis Rodr'iguez-Piazza
Abstract give estimates for the approximation numbers of composition operators on the Hp spaces, 1 ≤ p < ∞
摘要给出了1≤p <∞的Hp空间上复合算子的逼近数的估计
{"title":"Approximation numbers of composition operators on Hp","authors":"Daniel Li, Herv'e Queff'elec, Luis Rodr'iguez-Piazza","doi":"10.1515/conop-2015-0005","DOIUrl":"https://doi.org/10.1515/conop-2015-0005","url":null,"abstract":"Abstract give estimates for the approximation numbers of composition operators on the Hp spaces, 1 ≤ p < ∞","PeriodicalId":53800,"journal":{"name":"Concrete Operators","volume":"2 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2015-02-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/conop-2015-0005","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66887949","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 prove some results which give explicit methods for determining an upper bound for the rate of approximation by means of operators preserving a cone. Thenwe obtain some quantitative results on the rate of convergence for some sequences of linear shape-preserving operators.
{"title":"The Rate of Convergence for Linear Shape-Preserving Algorithms","authors":"Dmitry Boytsov, S. Sidorov","doi":"10.1515/conop-2015-0008","DOIUrl":"https://doi.org/10.1515/conop-2015-0008","url":null,"abstract":"Abstract We prove some results which give explicit methods for determining an upper bound for the rate of approximation by means of operators preserving a cone. Thenwe obtain some quantitative results on the rate of convergence for some sequences of linear shape-preserving operators.","PeriodicalId":53800,"journal":{"name":"Concrete Operators","volume":"2 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2015-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/conop-2015-0008","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66887683","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 the truncated operator trigonometric moment problem. All solutions of the moment problem are described by a Nevanlinna-type parameterization. In the case of moments acting in a separable Hilbert space, the matrices of the operator coefficients in the Nevanlinna-type formula are calculated by the prescribed moments. Conditions for the determinacy of the moment problem are given, as well.
{"title":"On the truncated operator trigonometric moment problem","authors":"S. Zagorodnyuk","doi":"10.1515/conop-2015-0002","DOIUrl":"https://doi.org/10.1515/conop-2015-0002","url":null,"abstract":"Abstract In this paper we study the truncated operator trigonometric moment problem. All solutions of the moment problem are described by a Nevanlinna-type parameterization. In the case of moments acting in a separable Hilbert space, the matrices of the operator coefficients in the Nevanlinna-type formula are calculated by the prescribed moments. Conditions for the determinacy of the moment problem are given, as well.","PeriodicalId":53800,"journal":{"name":"Concrete Operators","volume":"2 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2015-01-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/conop-2015-0002","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66887467","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 paper is a continuation of our previous investigations on the truncated matrix trigonometric moment problem in Ukrainian Math. J., 2011, 63, no. 6, 786-797, and Ukrainian Math. J., 2013, 64, no. 8, 1199- 1214. In this paper we shall study the truncated matrix trigonometric moment problem with an additional constraint posed on the matrix measure MT(δ), δ ∈ B(T), generated by the seeked function M(x): MT(∆) = 0, where ∆ is a given open subset of T (called a gap). We present necessary and sufficient conditions for the solvability of the moment problem with a gap. All solutions of the moment problem with a gap can be constructed by a Nevanlinna-type formula.
{"title":"The truncated matrix trigonometric moment problem with an open gap","authors":"S. Zagorodnyuk","doi":"10.2478/conop-2014-0003","DOIUrl":"https://doi.org/10.2478/conop-2014-0003","url":null,"abstract":"Abstract This paper is a continuation of our previous investigations on the truncated matrix trigonometric moment problem in Ukrainian Math. J., 2011, 63, no. 6, 786-797, and Ukrainian Math. J., 2013, 64, no. 8, 1199- 1214. In this paper we shall study the truncated matrix trigonometric moment problem with an additional constraint posed on the matrix measure MT(δ), δ ∈ B(T), generated by the seeked function M(x): MT(∆) = 0, where ∆ is a given open subset of T (called a gap). We present necessary and sufficient conditions for the solvability of the moment problem with a gap. All solutions of the moment problem with a gap can be constructed by a Nevanlinna-type formula.","PeriodicalId":53800,"journal":{"name":"Concrete Operators","volume":"30 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2014-03-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.2478/conop-2014-0003","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69192293","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 Given a symmetric linear transformation on a Hilbert space, a natural problem to consider is the characterization of its set of symmetric extensions. This problem is equivalent to the study of the partial isometric extensions of a fixed partial isometry. We provide a new function theoretic characterization of the set of all self-adjoint extensions of any symmetric linear transformation B with finite equal indices and inner Livšic characteristic function θB by constructing a bijection between the quotient of this set by a certain natural equivalence relation and the set of all contractive analytic functions φ which are greater or equal to θB.
{"title":"Extensions of symmetric operators I: The inner characteristic function case","authors":"R.T.W. Martin","doi":"10.1515/conop-2015-0004","DOIUrl":"https://doi.org/10.1515/conop-2015-0004","url":null,"abstract":"Abstract Given a symmetric linear transformation on a Hilbert space, a natural problem to consider is the characterization of its set of symmetric extensions. This problem is equivalent to the study of the partial isometric extensions of a fixed partial isometry. We provide a new function theoretic characterization of the set of all self-adjoint extensions of any symmetric linear transformation B with finite equal indices and inner Livšic characteristic function θB by constructing a bijection between the quotient of this set by a certain natural equivalence relation and the set of all contractive analytic functions φ which are greater or equal to θB.","PeriodicalId":53800,"journal":{"name":"Concrete Operators","volume":"2 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2014-03-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/conop-2015-0004","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66887154","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 describe the C*-algebra associated with the finite sections discretization of truncated Toeplitz operators on the model space K2u where u is an infinite Blaschke product. As consequences, we get a stability criterion for the finite sections discretization and results on spectral and pseudospectral approximation.
{"title":"Finite sections of truncated Toeplitz operators","authors":"S. Roch","doi":"10.2478/conop-2014-0002","DOIUrl":"https://doi.org/10.2478/conop-2014-0002","url":null,"abstract":"Abstract We describe the C*-algebra associated with the finite sections discretization of truncated Toeplitz operators on the model space K2u where u is an infinite Blaschke product. As consequences, we get a stability criterion for the finite sections discretization and results on spectral and pseudospectral approximation.","PeriodicalId":53800,"journal":{"name":"Concrete Operators","volume":"2 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2014-01-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.2478/conop-2014-0002","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69192729","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 the beginning of the twentieth century, Plemelj introduced the notion of factorization of matrix functions. The matrix factorization finds applications in many fields such as in the diffraction theory, in the theory of differential equations and in the theory of singular integral operators. However, the explicit formulas for the factors of the factorization are known only in a few classes of matrices. In the present paper we consider a new approach to obtain the factorization of a rational matrix function, relative to the unit circle. The constructed method is based on the relation between the general solution of a homogeneous Riemann-Hilbert problem and a solution of a linear system of difference equations with constant coefficients.
{"title":"Factorization of rational matrix functions and difference equations","authors":"J. S. Rodríguez, L. Campos","doi":"10.2478/conop-2012-0005","DOIUrl":"https://doi.org/10.2478/conop-2012-0005","url":null,"abstract":"Abstract In the beginning of the twentieth century, Plemelj introduced the notion of factorization of matrix functions. The matrix factorization finds applications in many fields such as in the diffraction theory, in the theory of differential equations and in the theory of singular integral operators. However, the explicit formulas for the factors of the factorization are known only in a few classes of matrices. In the present paper we consider a new approach to obtain the factorization of a rational matrix function, relative to the unit circle. The constructed method is based on the relation between the general solution of a homogeneous Riemann-Hilbert problem and a solution of a linear system of difference equations with constant coefficients.","PeriodicalId":53800,"journal":{"name":"Concrete Operators","volume":"1 1","pages":"37 - 53"},"PeriodicalIF":0.6,"publicationDate":"2013-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.2478/conop-2012-0005","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69192661","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 study the homogeneous Riemann-Hilbert problem with a vanishing scalar-valued continuous coefficient. We characterize non-existence of nontrivial solutions in the case where the coefficient has its values along several rays starting from the origin. As a consequence, some results on injectivity and existence of eigenvalues of Toeplitz operators in Hardy spaces are obtained.
{"title":"A Riemann-Hilbert problem with a vanishing coefficient and applications to Toeplitz operators","authors":"A. Perälä, J. Virtanen, L. Wolf","doi":"10.2478/conop-2012-0004","DOIUrl":"https://doi.org/10.2478/conop-2012-0004","url":null,"abstract":"Abstract We study the homogeneous Riemann-Hilbert problem with a vanishing scalar-valued continuous coefficient. We characterize non-existence of nontrivial solutions in the case where the coefficient has its values along several rays starting from the origin. As a consequence, some results on injectivity and existence of eigenvalues of Toeplitz operators in Hardy spaces are obtained.","PeriodicalId":53800,"journal":{"name":"Concrete Operators","volume":"1 1","pages":"28 - 36"},"PeriodicalIF":0.6,"publicationDate":"2013-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.2478/conop-2012-0004","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"69192644","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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