Abstract We present some recent results on Hardy spaces of generalized analytic functions on D specifying their link with the analytic Hardy spaces. Their definition can be extended to more general domains Ω . We discuss the way to extend such definitions to more general domains that depends on the regularity of the boundary of the domain ∂Ω. The generalization over general domains leads to the study of the invertibility of composition operators between Hardy spaces of generalized analytic functions; at the end of the paper, we discuss invertibility and Fredholm property of the composition operator C∅ on Hardy spaces of generalized analytic functions on a simply connected Dini-smooth domain for an analytic symbol ∅.
{"title":"Hardy spaces of generalized analytic functions and composition operators","authors":"Elodie Pozzi","doi":"10.1515/conop-2018-0002","DOIUrl":"https://doi.org/10.1515/conop-2018-0002","url":null,"abstract":"Abstract We present some recent results on Hardy spaces of generalized analytic functions on D specifying their link with the analytic Hardy spaces. Their definition can be extended to more general domains Ω . We discuss the way to extend such definitions to more general domains that depends on the regularity of the boundary of the domain ∂Ω. The generalization over general domains leads to the study of the invertibility of composition operators between Hardy spaces of generalized analytic functions; at the end of the paper, we discuss invertibility and Fredholm property of the composition operator C∅ on Hardy spaces of generalized analytic functions on a simply connected Dini-smooth domain for an analytic symbol ∅.","PeriodicalId":53800,"journal":{"name":"Concrete Operators","volume":"5 1","pages":"23 - 9"},"PeriodicalIF":0.6,"publicationDate":"2018-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/conop-2018-0002","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48672584","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 truncated multidimensional moment problem with a general type of truncations. The operator approach to the moment problem is presented. The case where the associated operators form a commuting self-adjoint tuple is characterized in terms of the given moments. The case of the dimensional stability is characterized in terms of the prescribed moments as well. Some sufficient conditions for the solvability of the moment problem are presented. A construction of the corresponding solution is described by algorithms. Numerical examples of the construction are provided.
{"title":"The operator approach to the truncated multidimensional moment problem","authors":"S. Zagorodnyuk","doi":"10.1515/conop-2019-0001","DOIUrl":"https://doi.org/10.1515/conop-2019-0001","url":null,"abstract":"Abstract We study the truncated multidimensional moment problem with a general type of truncations. The operator approach to the moment problem is presented. The case where the associated operators form a commuting self-adjoint tuple is characterized in terms of the given moments. The case of the dimensional stability is characterized in terms of the prescribed moments as well. Some sufficient conditions for the solvability of the moment problem are presented. A construction of the corresponding solution is described by algorithms. Numerical examples of the construction are provided.","PeriodicalId":53800,"journal":{"name":"Concrete Operators","volume":"6 1","pages":"1 - 19"},"PeriodicalIF":0.6,"publicationDate":"2018-02-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/conop-2019-0001","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43913209","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 Wiener-Hopf factorisation plays an important role in the theory of Toeplitz operators. We consider here Toeplitz operators in the Hardy spaces Hp of the upper half-plane and we review how their Fredholm properties can be studied in terms of a Wiener-Hopf factorisation of their symbols, obtaining necessary and sufficient conditions for the operator to be Fredholm or invertible, as well as formulae for their inverses or one-sided inverses when these exist. The results are applied to a class of singular integral equations in L−1(ℝ)
{"title":"Toeplitz operators and Wiener-Hopf factorisation: an introduction","authors":"M. Câmara","doi":"10.1515/conop-2017-0010","DOIUrl":"https://doi.org/10.1515/conop-2017-0010","url":null,"abstract":"Abstract Wiener-Hopf factorisation plays an important role in the theory of Toeplitz operators. We consider here Toeplitz operators in the Hardy spaces Hp of the upper half-plane and we review how their Fredholm properties can be studied in terms of a Wiener-Hopf factorisation of their symbols, obtaining necessary and sufficient conditions for the operator to be Fredholm or invertible, as well as formulae for their inverses or one-sided inverses when these exist. The results are applied to a class of singular integral equations in L−1(ℝ)","PeriodicalId":53800,"journal":{"name":"Concrete Operators","volume":"4 1","pages":"130 - 145"},"PeriodicalIF":0.6,"publicationDate":"2017-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/conop-2017-0010","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45905281","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 Being expected as a Banach space substitute of the orthogonal projections on Hilbert spaces, generalized n-circular projections also extend the notion of generalized bicontractive projections on JB*-triples. In this paper, we study some geometric properties of JB*-triples related to them. In particular, we provide some structure theorems of generalized n-circular projections on an often mentioned special case of JB*-triples, i.e., Hilbert C*-modules over abelian C*-algebras C0(Ω).
{"title":"Generalized n-circular projections on JB*-triples and Hilbert C0(Ω)-modules","authors":"D. Ilišević, Chih-Neng Liu, N. Wong","doi":"10.1515/conop-2017-0008","DOIUrl":"https://doi.org/10.1515/conop-2017-0008","url":null,"abstract":"Abstract Being expected as a Banach space substitute of the orthogonal projections on Hilbert spaces, generalized n-circular projections also extend the notion of generalized bicontractive projections on JB*-triples. In this paper, we study some geometric properties of JB*-triples related to them. In particular, we provide some structure theorems of generalized n-circular projections on an often mentioned special case of JB*-triples, i.e., Hilbert C*-modules over abelian C*-algebras C0(Ω).","PeriodicalId":53800,"journal":{"name":"Concrete Operators","volume":"4 1","pages":"109 - 120"},"PeriodicalIF":0.6,"publicationDate":"2017-10-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/conop-2017-0008","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43474928","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 selective survey on the space lAp and its multipliers. It also includes some connections of multipliers to Birkhoff-James orthogonality
{"title":"Multipliers of sequence spaces","authors":"R. Cheng, J. Mashreghi, W. Ross","doi":"10.1515/conop-2017-0007","DOIUrl":"https://doi.org/10.1515/conop-2017-0007","url":null,"abstract":"Abstract This paper is selective survey on the space lAp and its multipliers. It also includes some connections of multipliers to Birkhoff-James orthogonality","PeriodicalId":53800,"journal":{"name":"Concrete Operators","volume":"4 1","pages":"108 - 76"},"PeriodicalIF":0.6,"publicationDate":"2017-10-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/conop-2017-0007","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45310094","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 Hilbert spaces ℋw consisiting of Dirichlet series F(s)=∑n=1∞ann-s $F(s) = sumnolimits_{n = 1}^infty {{a_n}{n^{ - s}}}$ that satisfty ∑n=1∞|an|2/wn<∞ ${sumnolimits_{n = 1}^infty {left| {{a_n}} right|} ^2}/{w_n} < infty $ with {wn}n of average order logj n (the j-fold logarithm of n), can be embedded into certain small Bergman spaces. Using this embedding, we study the Gordon–Hedenmalm theorem on such ℋw from an iterative point of view. By that theorem, the composition operators are generated by functions of the form Φ (s) = c0s +ϕ(s), where c0 is a nonnegative integer and ϕ is a Dirichlet series with certain convergence and mapping properties. The iterative phenomenon takes place when c0 = 0. It is verified for every integer j ⩾ 1, real α > 0 and {wn}n having average order (logj+n)α ${(log _j^ + n)^alpha }$ , that the composition operators map ℋw into a scale of ℋw’ with w’n having average order (logj+1+n)α ${(log _{j + 1}^ + n)^alpha }$ . The case j = 1 can be deduced from the proof of the main theorem of a recent paper of Bailleul and Brevig, and we adopt the same method to study the general iterative step.
{"title":"Iteration of Composition Operators on small Bergman spaces of Dirichlet series","authors":"J. Zhao","doi":"10.1515/conop-2018-0003","DOIUrl":"https://doi.org/10.1515/conop-2018-0003","url":null,"abstract":"Abstract The Hilbert spaces ℋw consisiting of Dirichlet series F(s)=∑n=1∞ann-s $F(s) = sumnolimits_{n = 1}^infty {{a_n}{n^{ - s}}}$ that satisfty ∑n=1∞|an|2/wn<∞ ${sumnolimits_{n = 1}^infty {left| {{a_n}} right|} ^2}/{w_n} < infty $ with {wn}n of average order logj n (the j-fold logarithm of n), can be embedded into certain small Bergman spaces. Using this embedding, we study the Gordon–Hedenmalm theorem on such ℋw from an iterative point of view. By that theorem, the composition operators are generated by functions of the form Φ (s) = c0s +ϕ(s), where c0 is a nonnegative integer and ϕ is a Dirichlet series with certain convergence and mapping properties. The iterative phenomenon takes place when c0 = 0. It is verified for every integer j ⩾ 1, real α > 0 and {wn}n having average order (logj+n)α ${(log _j^ + n)^alpha }$ , that the composition operators map ℋw into a scale of ℋw’ with w’n having average order (logj+1+n)α ${(log _{j + 1}^ + n)^alpha }$ . The case j = 1 can be deduced from the proof of the main theorem of a recent paper of Bailleul and Brevig, and we adopt the same method to study the general iterative step.","PeriodicalId":53800,"journal":{"name":"Concrete Operators","volume":"5 1","pages":"24 - 34"},"PeriodicalIF":0.6,"publicationDate":"2017-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/conop-2018-0003","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42479959","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}
I. Louhichi, Fanilo Randriamahaleo, Lova Zakariasy
Abstract One of the major goals in the theory of Toeplitz operators on the Bergman space over the unit disk D in the complex place C is to completely describe the commutant of a given Toeplitz operator, that is, the set of all Toeplitz operators that commute with it. Here we shall study the commutants of a certain class of quasihomogeneous Toeplitz operators defined on the harmonic Bergman space.
{"title":"On the Commutativity of a Certain Class of Toeplitz Operators","authors":"I. Louhichi, Fanilo Randriamahaleo, Lova Zakariasy","doi":"10.2478/conop-2014-0001","DOIUrl":"https://doi.org/10.2478/conop-2014-0001","url":null,"abstract":"Abstract One of the major goals in the theory of Toeplitz operators on the Bergman space over the unit disk D in the complex place C is to completely describe the commutant of a given Toeplitz operator, that is, the set of all Toeplitz operators that commute with it. Here we shall study the commutants of a certain class of quasihomogeneous Toeplitz operators defined on the harmonic Bergman space.","PeriodicalId":53800,"journal":{"name":"Concrete Operators","volume":"2 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2017-04-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.2478/conop-2014-0001","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44894679","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 Drury-Arveson space, we consider the subspace of functions whose Taylor coefficients are supported in a set Y⊂ ℕd with the property that ℕX + ej ⊂ ℕX for all j = 1, . . . , d. This is an easy example of shift-invariant subspace, which can be considered as a RKHS in is own right, with a kernel that can be explicitly calculated for specific choices of X. Every such a space can be seen as an intersection of kernels of Hankel operators with explicit symbols. Finally, this is the right space on which Drury’s inequality can be optimally adapted to a sub-family of the commuting and contractive operators originally considered by Drury.
{"title":"On a class of shift-invariant subspaces of the Drury-Arveson space","authors":"N. Arcozzi, Matteo Levi","doi":"10.1515/conop-2018-0001","DOIUrl":"https://doi.org/10.1515/conop-2018-0001","url":null,"abstract":"Abstract In the Drury-Arveson space, we consider the subspace of functions whose Taylor coefficients are supported in a set Y⊂ ℕd with the property that ℕX + ej ⊂ ℕX for all j = 1, . . . , d. This is an easy example of shift-invariant subspace, which can be considered as a RKHS in is own right, with a kernel that can be explicitly calculated for specific choices of X. Every such a space can be seen as an intersection of kernels of Hankel operators with explicit symbols. Finally, this is the right space on which Drury’s inequality can be optimally adapted to a sub-family of the commuting and contractive operators originally considered by Drury.","PeriodicalId":53800,"journal":{"name":"Concrete Operators","volume":"5 1","pages":"1 - 8"},"PeriodicalIF":0.6,"publicationDate":"2017-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/conop-2018-0001","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46155821","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 Using the Kato-Rosenblum theorem, we describe the absolutely continuous spectrum of a class of weighted integral Hankel operators in L2(ℝ+). These self-adjoint operators generalise the explicitly diagonalisable operator with the integral kernel sαtα(s + t)-1-2α, where α > -1/2. Our analysis can be considered as an extension of J. Howland’s 1992 paper which dealt with the unweighted case, corresponding to α = 0.
{"title":"Weighted integral Hankel operators with continuous spectrum","authors":"Emilio Fedele, A. Pushnitski","doi":"10.1515/conop-2017-0009","DOIUrl":"https://doi.org/10.1515/conop-2017-0009","url":null,"abstract":"Abstract Using the Kato-Rosenblum theorem, we describe the absolutely continuous spectrum of a class of weighted integral Hankel operators in L2(ℝ+). These self-adjoint operators generalise the explicitly diagonalisable operator with the integral kernel sαtα(s + t)-1-2α, where α > -1/2. Our analysis can be considered as an extension of J. Howland’s 1992 paper which dealt with the unweighted case, corresponding to α = 0.","PeriodicalId":53800,"journal":{"name":"Concrete Operators","volume":"4 1","pages":"121 - 129"},"PeriodicalIF":0.6,"publicationDate":"2017-02-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/conop-2017-0009","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48230471","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 consider the infinite Jacobi block matrices in the completely indeterminate case, i. e. such that the deficiency indices of the corresponding Jacobi operators are maximal. For such matrices, some criteria of complete indeterminacy are established. These criteria are similar to several known criteria of indeterminacy of the Hamburger moment problem in terms of the corresponding scalar Jacobi matrices and the related systems of orthogonal polynomials.
{"title":"On the completely indeterminate case for block Jacobi matrices","authors":"A. Osipov","doi":"10.1515/conop-2017-0005","DOIUrl":"https://doi.org/10.1515/conop-2017-0005","url":null,"abstract":"Abstract We consider the infinite Jacobi block matrices in the completely indeterminate case, i. e. such that the deficiency indices of the corresponding Jacobi operators are maximal. For such matrices, some criteria of complete indeterminacy are established. These criteria are similar to several known criteria of indeterminacy of the Hamburger moment problem in terms of the corresponding scalar Jacobi matrices and the related systems of orthogonal polynomials.","PeriodicalId":53800,"journal":{"name":"Concrete Operators","volume":"4 1","pages":"48 - 57"},"PeriodicalIF":0.6,"publicationDate":"2017-01-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/conop-2017-0005","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49357846","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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