Abstract Given Banach spaces X and Y, we ask about the dual space of the (X, Y). This paper surveys results on tensor products of Banach spaces with the main objective of describing the dual of spaces of bounded operators. In several cases and under a variety of assumptions on X and Y, the answer can best be given as the projective tensor product of X** and Y*.
{"title":"The dual of the space of bounded operators on a Banach space","authors":"F. Botelho, R. Fleming","doi":"10.1515/conop-2020-0109","DOIUrl":"https://doi.org/10.1515/conop-2020-0109","url":null,"abstract":"Abstract Given Banach spaces X and Y, we ask about the dual space of the (X, Y). This paper surveys results on tensor products of Banach spaces with the main objective of describing the dual of spaces of bounded operators. In several cases and under a variety of assumptions on X and Y, the answer can best be given as the projective tensor product of X** and Y*.","PeriodicalId":53800,"journal":{"name":"Concrete Operators","volume":"8 1","pages":"48 - 59"},"PeriodicalIF":0.6,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/conop-2020-0109","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45255468","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}
B. Venkateswarlu, P. Reddy, R. M. Shilpa, G. Swapna
Abstract In this paper,we introduce and study a new subclass of meromorphic functions associated with a certain differential operator on Hilbert space. For this class, we obtain several properties like the coefficient inequality, growth and distortion theorem, radius of close-to-convexity, starlikeness and meromorphically convexity and integral transforms. Further, it is shown that this class is closed under convex linear combinations.
{"title":"A Note on Meromorphic Functions Associated With Beseel Function Defined by Hilbert Sapce Operator","authors":"B. Venkateswarlu, P. Reddy, R. M. Shilpa, G. Swapna","doi":"10.1515/conop-2020-0114","DOIUrl":"https://doi.org/10.1515/conop-2020-0114","url":null,"abstract":"Abstract In this paper,we introduce and study a new subclass of meromorphic functions associated with a certain differential operator on Hilbert space. For this class, we obtain several properties like the coefficient inequality, growth and distortion theorem, radius of close-to-convexity, starlikeness and meromorphically convexity and integral transforms. Further, it is shown that this class is closed under convex linear combinations.","PeriodicalId":53800,"journal":{"name":"Concrete Operators","volume":"8 1","pages":"66 - 76"},"PeriodicalIF":0.6,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/conop-2020-0114","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45072970","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 investigate the numerical range of C*bφm Caφn and Caφn C*bφm on the Hardy space where φ is an inner function fixing the origin and a and b are points in the open unit disc. In the case when |a| = |b| = 1 we characterize the numerical range of these operators by constructing lacunary polynomials of unit norm whose image under the quadratic form incrementally foliate the numerical range. In the case when a and b are small we show numerical range of both operators is equal to the numerical range of the operator restricted to a 3-dimensional subspace.
{"title":"The Numerical Range of C*ψ Cφ and Cφ C*ψ","authors":"J. Clifford, Michael Dabkowski, Alan D. Wiggins","doi":"10.1515/conop-2020-0108","DOIUrl":"https://doi.org/10.1515/conop-2020-0108","url":null,"abstract":"Abstract In this paper we investigate the numerical range of C*bφm Caφn and Caφn C*bφm on the Hardy space where φ is an inner function fixing the origin and a and b are points in the open unit disc. In the case when |a| = |b| = 1 we characterize the numerical range of these operators by constructing lacunary polynomials of unit norm whose image under the quadratic form incrementally foliate the numerical range. In the case when a and b are small we show numerical range of both operators is equal to the numerical range of the operator restricted to a 3-dimensional subspace.","PeriodicalId":53800,"journal":{"name":"Concrete Operators","volume":"8 1","pages":"13 - 23"},"PeriodicalIF":0.6,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/conop-2020-0108","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44425844","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 For 1 < p < ∞, the Besov space Bp consists of those functions f which are analytic in the unit disc 𝔻 = {z ∈ : |z| < 1} and satisfy ∫𝔻(1 − |z|2)p−2|f ′(z)|p dA(z) < ∞. The space B2 reduces to the classical Dirichlet space 𝒟. It is known that if f ∈ 𝒟then |f ′(reiθ)| = o[(1 − r)−1/2], for almost every ∈ [0, 2π]. Hallenbeck and Samotij proved that this result is sharp in a very strong sense. We obtain substitutes of the above results valid for the spaces Bp (1 < p < ∞) an we give also an application of our them to questions concerning multipliers between Besov spaces.
{"title":"Radial growth of the derivatives of analytic functions in Besov spaces","authors":"S. Dominguez, D. Girela","doi":"10.1515/conop-2020-0107","DOIUrl":"https://doi.org/10.1515/conop-2020-0107","url":null,"abstract":"Abstract For 1 < p < ∞, the Besov space Bp consists of those functions f which are analytic in the unit disc 𝔻 = {z ∈ : |z| < 1} and satisfy ∫𝔻(1 − |z|2)p−2|f ′(z)|p dA(z) < ∞. The space B2 reduces to the classical Dirichlet space 𝒟. It is known that if f ∈ 𝒟then |f ′(reiθ)| = o[(1 − r)−1/2], for almost every ∈ [0, 2π]. Hallenbeck and Samotij proved that this result is sharp in a very strong sense. We obtain substitutes of the above results valid for the spaces Bp (1 < p < ∞) an we give also an application of our them to questions concerning multipliers between Besov spaces.","PeriodicalId":53800,"journal":{"name":"Concrete Operators","volume":"8 1","pages":"1 - 12"},"PeriodicalIF":0.6,"publicationDate":"2020-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/conop-2020-0107","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41538394","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 cyclic, supercyclic and hypercyclic properties of a composition operator Cϕ on the Segal-Bargmann space ℋ(ℰ), where ϕ(z) = Az + b, A is a bounded linear operator on ℰ, b ∈ ℰ with ||A|| ⩽ 1 and A*b belongs to the range of (I – A*A)½. Specifically, under some conditions on the symbol ϕ we show that if Cϕ is cyclic then A* is cyclic but the converse need not be true. We also show that if Cϕ* is cyclic then A is cyclic. Further we show that there is no supercyclic composition operator on the space ℋ(ℰ) for certain class of symbols ϕ.
{"title":"Cyclic Composition operators on Segal-Bargmann space","authors":"G. Ramesh, B. S. Ranjan, D. Naidu","doi":"10.1515/conop-2022-0133","DOIUrl":"https://doi.org/10.1515/conop-2022-0133","url":null,"abstract":"Abstract We study the cyclic, supercyclic and hypercyclic properties of a composition operator Cϕ on the Segal-Bargmann space ℋ(ℰ), where ϕ(z) = Az + b, A is a bounded linear operator on ℰ, b ∈ ℰ with ||A|| ⩽ 1 and A*b belongs to the range of (I – A*A)½. Specifically, under some conditions on the symbol ϕ we show that if Cϕ is cyclic then A* is cyclic but the converse need not be true. We also show that if Cϕ* is cyclic then A is cyclic. Further we show that there is no supercyclic composition operator on the space ℋ(ℰ) for certain class of symbols ϕ.","PeriodicalId":53800,"journal":{"name":"Concrete Operators","volume":"9 1","pages":"127 - 138"},"PeriodicalIF":0.6,"publicationDate":"2020-11-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45511430","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 main purpose of this work is the construction of an analytic functional calculus for Clifford operators, which are operators acting on certain modules over Clifford algebras. Unlike in some preceding works by other authors, we use a spectrum defined in the complex plane, and also certain stem functions, analytic in neighborhoods of such a spectrum. The replacement of the slice regular functions, having values in a Clifford algebra, by analytic stem functions becomes possible because of an isomorphism induced by a Cauchy type transform, whose existence is proved in the first part of this work.
{"title":"Spectrum and Analytic Functional Calculus for Clifford Operators via Stem Functions","authors":"F. Vasilescu","doi":"10.1515/conop-2020-0115","DOIUrl":"https://doi.org/10.1515/conop-2020-0115","url":null,"abstract":"Abstract The main purpose of this work is the construction of an analytic functional calculus for Clifford operators, which are operators acting on certain modules over Clifford algebras. Unlike in some preceding works by other authors, we use a spectrum defined in the complex plane, and also certain stem functions, analytic in neighborhoods of such a spectrum. The replacement of the slice regular functions, having values in a Clifford algebra, by analytic stem functions becomes possible because of an isomorphism induced by a Cauchy type transform, whose existence is proved in the first part of this work.","PeriodicalId":53800,"journal":{"name":"Concrete Operators","volume":"8 1","pages":"90 - 113"},"PeriodicalIF":0.6,"publicationDate":"2020-08-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/conop-2020-0115","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45133726","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 prove our main result that the Li-Yorke chaotic eigen set of a positive integer multiple of the backward shift operator on ℓ2 () is a disk in the complex plane and the union of such Li-Yorke chaotic eigen set’s is the whole complex plane .
{"title":"Li-Yorke chaotic eigen set of the backward shift operator on ℓ2()","authors":"B. Sanooj, P. Vinodkumar","doi":"10.1515/conop-2020-0105","DOIUrl":"https://doi.org/10.1515/conop-2020-0105","url":null,"abstract":"Abstract In this paper, we prove our main result that the Li-Yorke chaotic eigen set of a positive integer multiple of the backward shift operator on ℓ2 () is a disk in the complex plane and the union of such Li-Yorke chaotic eigen set’s is the whole complex plane .","PeriodicalId":53800,"journal":{"name":"Concrete Operators","volume":"7 1","pages":"180 - 182"},"PeriodicalIF":0.6,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/conop-2020-0105","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48470400","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 Paley Wiener theorem characterizes the class of functions which are Fourier transforms of ℂ∞ functions of compact support on ℝn by relating decay properties of those functions or distributions at infinity with analyticity of their Fourier transform. The theorem is already proved in classical case : the real case with holomorphic Fourier transform on L2(ℝ), the case of functions with compact support on ℝn from Hörmander and the spherical transform on semi simple Lie groups with Gangolli theorem. Let G be a locally compact unimodular group, K a compact subgroup of G, and δ an element of unitary dual ̑K of K. In this work, we’ll give an extension of Paley-Wiener theorem with respect to δ, a class of unitary irreducible representation of K, where G is either a semi-simple Lie group or a reductive Lie group with nonempty discrete series after introducing a notion of δ-orbital integral. If δ is trivial and one dimensional, we obtain the classical Paley-Wiener theorem.
{"title":"On some extension of Paley Wiener theorem","authors":"Ettien Yves-Fernand N’Da, K. Kangni","doi":"10.1515/conop-2020-0006","DOIUrl":"https://doi.org/10.1515/conop-2020-0006","url":null,"abstract":"Abstract Paley Wiener theorem characterizes the class of functions which are Fourier transforms of ℂ∞ functions of compact support on ℝn by relating decay properties of those functions or distributions at infinity with analyticity of their Fourier transform. The theorem is already proved in classical case : the real case with holomorphic Fourier transform on L2(ℝ), the case of functions with compact support on ℝn from Hörmander and the spherical transform on semi simple Lie groups with Gangolli theorem. Let G be a locally compact unimodular group, K a compact subgroup of G, and δ an element of unitary dual ̑K of K. In this work, we’ll give an extension of Paley-Wiener theorem with respect to δ, a class of unitary irreducible representation of K, where G is either a semi-simple Lie group or a reductive Lie group with nonempty discrete series after introducing a notion of δ-orbital integral. If δ is trivial and one dimensional, we obtain the classical Paley-Wiener theorem.","PeriodicalId":53800,"journal":{"name":"Concrete Operators","volume":"7 1","pages":"81 - 90"},"PeriodicalIF":0.6,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/conop-2020-0006","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47017767","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 compactness and the hypercyclicity of Toeplitz operators Tϕ¯,b {T_{bar varphi ,b}} with co-analytic and bounded symbols on de Branges-Rovnyak spaces ℋ(b). For the compactness of Tϕ¯,b {T_{bar varphi ,b}} , we will see that the result depends on the boundary spectrum of b. We will prove that there are non trivial compact operators of the form Tϕ¯,b {T_{bar varphi ,b}} , with ϕ ∈ H∞ ∩ C(𝕋), if and only if m(σ(b) ∩ 𝕋) = 0. We will also show that, when b is non-extreme, then Tϕ¯,b {T_{bar varphi ,b}} is hypercyclic if and only if ϕ is non-constant and ϕ(𝔻) ∩ 𝕋 ≠ ∅.
{"title":"Compactness and hypercyclicity of co-analytic Toeplitz operators on de Branges-Rovnyak spaces","authors":"Rim Alhajj","doi":"10.1515/conop-2020-0004","DOIUrl":"https://doi.org/10.1515/conop-2020-0004","url":null,"abstract":"Abstract We study the compactness and the hypercyclicity of Toeplitz operators Tϕ¯,b {T_{bar varphi ,b}} with co-analytic and bounded symbols on de Branges-Rovnyak spaces ℋ(b). For the compactness of Tϕ¯,b {T_{bar varphi ,b}} , we will see that the result depends on the boundary spectrum of b. We will prove that there are non trivial compact operators of the form Tϕ¯,b {T_{bar varphi ,b}} , with ϕ ∈ H∞ ∩ C(𝕋), if and only if m(σ(b) ∩ 𝕋) = 0. We will also show that, when b is non-extreme, then Tϕ¯,b {T_{bar varphi ,b}} is hypercyclic if and only if ϕ is non-constant and ϕ(𝔻) ∩ 𝕋 ≠ ∅.","PeriodicalId":53800,"journal":{"name":"Concrete Operators","volume":"7 1","pages":"55 - 68"},"PeriodicalIF":0.6,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/conop-2020-0004","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46226868","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 tools of Sz.-Nagy–Foias theory of contractions, we describe in detail the invariant subspaces of the operator S ⊕ S*, where S is the unilateral shift on a Hilbert space. This answers a question of Câmara and Ross.
{"title":"The invariant subspaces of S ⊕ S*","authors":"D. Timotin","doi":"10.1515/conop-2020-0101","DOIUrl":"https://doi.org/10.1515/conop-2020-0101","url":null,"abstract":"Abstract Using the tools of Sz.-Nagy–Foias theory of contractions, we describe in detail the invariant subspaces of the operator S ⊕ S*, where S is the unilateral shift on a Hilbert space. This answers a question of Câmara and Ross.","PeriodicalId":53800,"journal":{"name":"Concrete Operators","volume":"7 1","pages":"116 - 123"},"PeriodicalIF":0.6,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/conop-2020-0101","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45587602","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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