Abstract We study a Miura-type transformation between Kac - van Moerbeke (Volterra) and Toda lattices in terms of the inverse spectral problem for Jacobi operators, which appear in the Lax representation for such systems. This inverse problem method, which amounts to reconstruction of the operator from the moments of its Weyl function, can be used in solving initial-boundary value problem for both systems. It is shown that the Miura transformation can be easily described in terms of these moments. Using this description we establish a bijection between the Volterra lattices and the class of Toda lattices which is characterized by positivity of Jacobi operators in their Lax representation. Also, we discuss an implication of the latter result to the spectral theory.
{"title":"Inverse spectral problem for Jacobi operators and Miura transformation","authors":"A. Osipov","doi":"10.1515/conop-2020-0116","DOIUrl":"https://doi.org/10.1515/conop-2020-0116","url":null,"abstract":"Abstract We study a Miura-type transformation between Kac - van Moerbeke (Volterra) and Toda lattices in terms of the inverse spectral problem for Jacobi operators, which appear in the Lax representation for such systems. This inverse problem method, which amounts to reconstruction of the operator from the moments of its Weyl function, can be used in solving initial-boundary value problem for both systems. It is shown that the Miura transformation can be easily described in terms of these moments. Using this description we establish a bijection between the Volterra lattices and the class of Toda lattices which is characterized by positivity of Jacobi operators in their Lax representation. Also, we discuss an implication of the latter result to the spectral theory.","PeriodicalId":53800,"journal":{"name":"Concrete Operators","volume":"11 4","pages":"77 - 89"},"PeriodicalIF":0.6,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/conop-2020-0116","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41299699","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 shall introduce functions spaces as subspaces of Lpκ (ℝ) that we call Besov-κ-Hankel spaces and extend the concept of κ-Hankel wavelet transform in Lpκ(ℝ) space. Subsequently we will characterize the Besov-κ-Hankel space by using κ-Hankel wavelet coefficients.
{"title":"Besov-type spaces for the κ-Hankel wavelet transform on the real line","authors":"Ashish Pathak, Shrish Pandey","doi":"10.1515/conop-2020-0117","DOIUrl":"https://doi.org/10.1515/conop-2020-0117","url":null,"abstract":"Abstract In this paper, we shall introduce functions spaces as subspaces of Lpκ (ℝ) that we call Besov-κ-Hankel spaces and extend the concept of κ-Hankel wavelet transform in Lpκ(ℝ) space. Subsequently we will characterize the Besov-κ-Hankel space by using κ-Hankel wavelet coefficients.","PeriodicalId":53800,"journal":{"name":"Concrete Operators","volume":"8 1","pages":"114 - 124"},"PeriodicalIF":0.6,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44630926","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 {S(t)}t≥ 0 be an integrated semigroup of bounded linear operators on the Banach space 𝒳 into itself and let A be their generator. In this paper, we consider some necessary and sufficient conditions for the Cesàro mean and the Abel average of S(t) converge uniformly on ℬ(𝒳). More precisely, we show that the Abel average of S(t) converges uniformly if and only if 𝒳 = ℛ(A) ⊕ 𝒩(A), if and only if ℛ(Ak) is closed for some integer k and ∥ λ2R(λ, A) ∥ → 0 as λ→ 0+, where ℛ(A), 𝒩(A) and R(λ, A), be the range, the kernel, the resolvent function of A, respectively. Furthermore, we prove that if S(t)/t2 → 0 as t → 1, then the Cesàro mean of S(t) converges uniformly if and only if the Abel average of S(t) is also converges uniformly.
{"title":"Cesàro and Abel ergodic theorems for integrated semigroups","authors":"F. Barki","doi":"10.1515/conop-2020-0119","DOIUrl":"https://doi.org/10.1515/conop-2020-0119","url":null,"abstract":"Abstract Let {S(t)}t≥ 0 be an integrated semigroup of bounded linear operators on the Banach space 𝒳 into itself and let A be their generator. In this paper, we consider some necessary and sufficient conditions for the Cesàro mean and the Abel average of S(t) converge uniformly on ℬ(𝒳). More precisely, we show that the Abel average of S(t) converges uniformly if and only if 𝒳 = ℛ(A) ⊕ 𝒩(A), if and only if ℛ(Ak) is closed for some integer k and ∥ λ2R(λ, A) ∥ → 0 as λ→ 0+, where ℛ(A), 𝒩(A) and R(λ, A), be the range, the kernel, the resolvent function of A, respectively. Furthermore, we prove that if S(t)/t2 → 0 as t → 1, then the Cesàro mean of S(t) converges uniformly if and only if the Abel average of S(t) is also converges uniformly.","PeriodicalId":53800,"journal":{"name":"Concrete Operators","volume":"8 1","pages":"135 - 149"},"PeriodicalIF":0.6,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45087513","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 research, we introduce a new fractional Cesàro matrix and investigate the topological properties of the sequence space associated with this matrix.We also introduce a fractional Gamma matrix aswell and obtain some factorizations for the Hilbert operator based on Cesàro and Gamma matrices. The results of these factorizations are two new inequalities one ofwhich is a generalized version of thewell-known Hilbert’s inequality. There are also some challenging problems that authors share at the end of the manuscript and invite the researcher for trying to solve them.
{"title":"Fractional Cesàro Matrix and its Associated Sequence Space","authors":"H. Roopaei, M. İlkhan","doi":"10.1515/conop-2020-0112","DOIUrl":"https://doi.org/10.1515/conop-2020-0112","url":null,"abstract":"Abstract In this research, we introduce a new fractional Cesàro matrix and investigate the topological properties of the sequence space associated with this matrix.We also introduce a fractional Gamma matrix aswell and obtain some factorizations for the Hilbert operator based on Cesàro and Gamma matrices. The results of these factorizations are two new inequalities one ofwhich is a generalized version of thewell-known Hilbert’s inequality. There are also some challenging problems that authors share at the end of the manuscript and invite the researcher for trying to solve them.","PeriodicalId":53800,"journal":{"name":"Concrete Operators","volume":"8 1","pages":"24 - 39"},"PeriodicalIF":0.6,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/conop-2020-0112","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43279838","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 = (Tt)t≥0 be a C0-semigroup on a separable infinite dimensional Banach space X, with generator A. In this paper, we study the relationship between the single valued extension property of generator A, and the M-hypercyclicity of the C0-semigroup. Specifically, we prove that if A does not have the single valued extension property at λ ∈ iℝ, then there exists a closed subspace M of X, such that the C0-semigroup 𝒯 is M-hypercyclic. As a corollary, we get certain conditions of the generator A, for the C0-semigroup to be M-hypercyclic.
{"title":"M-hypercyclicity of C0-semigroup and Svep of its generator","authors":"A. Toukmati","doi":"10.1515/conop-2020-0122","DOIUrl":"https://doi.org/10.1515/conop-2020-0122","url":null,"abstract":"Abstract Let 𝒯 = (Tt)t≥0 be a C0-semigroup on a separable infinite dimensional Banach space X, with generator A. In this paper, we study the relationship between the single valued extension property of generator A, and the M-hypercyclicity of the C0-semigroup. Specifically, we prove that if A does not have the single valued extension property at λ ∈ iℝ, then there exists a closed subspace M of X, such that the C0-semigroup 𝒯 is M-hypercyclic. As a corollary, we get certain conditions of the generator A, for the C0-semigroup to be M-hypercyclic.","PeriodicalId":53800,"journal":{"name":"Concrete Operators","volume":"8 1","pages":"187 - 191"},"PeriodicalIF":0.6,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44091398","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 define a conformable diff-integral operator for a class of meromorphically multivalent functions. We show that this conformable operator adheres to the semigroup property. We then use the subordination properties to prove inclusion conditions, sufficienrt inclusion conditions and convolution properties for this class of conformable operators.
{"title":"Conformable differential operators for meromorphically multivalent functions","authors":"R. Ibrahim, D. Baleanu, J. Jahangiri","doi":"10.1515/conop-2020-0113","DOIUrl":"https://doi.org/10.1515/conop-2020-0113","url":null,"abstract":"Abstract We define a conformable diff-integral operator for a class of meromorphically multivalent functions. We show that this conformable operator adheres to the semigroup property. We then use the subordination properties to prove inclusion conditions, sufficienrt inclusion conditions and convolution properties for this class of conformable operators.","PeriodicalId":53800,"journal":{"name":"Concrete Operators","volume":"8 1","pages":"150 - 157"},"PeriodicalIF":0.6,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45920403","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 generalisation of m-expansive Hilbert space operators T ∈ B(ℋ) [18, 20] to Banach space operators T ∈ B(𝒳) is obtained by defining that a pair of operators A, B ∈ B(𝒳) is (m, P)-expansive for some operator P ∈ B(𝒳) if Δ A,Bm(P)= (I-LARB)m(P)=∑j=0m(-1)j(jm){left( {I - {L_A}{R_B}} right)^m}left( P right) = sumnolimits_{j = 0}^m {{{left( { - 1} right)}^j}left( {_j^m} right)}AjPBj≤0; LA(X) = AX and RB(X)=XB. Unlike m-isometric and m-left invertible operators, commuting products and perturbations by commuting nilpotents of (m, I)-expansive operators do not result in expansive operators: using elementary algebraic properties of the left and right multiplication operators, a sufficient condition is proved. For Drazin invertible A and B ∈ B(ℋ), with Drazin inverses Ad and Bd, a sufficient condition proving (Ad, Bd) ^ (A, B) is (m − 1, P)-isometric (resp., (m − 1, P)-contractive) for m even (resp., m odd) is given, and a Banach space analogue of this result is proved.
{"title":"On (m, P)-expansive operators: products, perturbation by nilpotents, Drazin invertibility","authors":"B. Duggal","doi":"10.1515/conop-2020-0120","DOIUrl":"https://doi.org/10.1515/conop-2020-0120","url":null,"abstract":"Abstract A generalisation of m-expansive Hilbert space operators T ∈ B(ℋ) [18, 20] to Banach space operators T ∈ B(𝒳) is obtained by defining that a pair of operators A, B ∈ B(𝒳) is (m, P)-expansive for some operator P ∈ B(𝒳) if Δ A,Bm(P)= (I-LARB)m(P)=∑j=0m(-1)j(jm){left( {I - {L_A}{R_B}} right)^m}left( P right) = sumnolimits_{j = 0}^m {{{left( { - 1} right)}^j}left( {_j^m} right)}AjPBj≤0; LA(X) = AX and RB(X)=XB. Unlike m-isometric and m-left invertible operators, commuting products and perturbations by commuting nilpotents of (m, I)-expansive operators do not result in expansive operators: using elementary algebraic properties of the left and right multiplication operators, a sufficient condition is proved. For Drazin invertible A and B ∈ B(ℋ), with Drazin inverses Ad and Bd, a sufficient condition proving (Ad, Bd) ^ (A, B) is (m − 1, P)-isometric (resp., (m − 1, P)-contractive) for m even (resp., m odd) is given, and a Banach space analogue of this result is proved.","PeriodicalId":53800,"journal":{"name":"Concrete Operators","volume":"8 1","pages":"158 - 175"},"PeriodicalIF":0.6,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41572229","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 (C(t))t∈ℝ be a strongly continuous cosine family and A be its infinitesimal generator. In this work, we prove that, if C(t) – cosh λt is semi-Fredholm (resp. semi-Browder, Drazin inversible, left essentially Drazin and right essentially Drazin invertible) operator and λt ∉ iπℤ, then A – λ2 is also. We show by counterexample that the converse is false in general.
{"title":"Spectral Theory For Strongly Continuous Cosine","authors":"H. Boua","doi":"10.1515/conop-2020-0110","DOIUrl":"https://doi.org/10.1515/conop-2020-0110","url":null,"abstract":"Abstract Let (C(t))t∈ℝ be a strongly continuous cosine family and A be its infinitesimal generator. In this work, we prove that, if C(t) – cosh λt is semi-Fredholm (resp. semi-Browder, Drazin inversible, left essentially Drazin and right essentially Drazin invertible) operator and λt ∉ iπℤ, then A – λ2 is also. We show by counterexample that the converse is false in general.","PeriodicalId":53800,"journal":{"name":"Concrete Operators","volume":"8 1","pages":"40 - 47"},"PeriodicalIF":0.6,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/conop-2020-0110","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45549082","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 recent work derived expressions for the induced p-norm of a special class of circulant matrices A(n, a, b) ∈ ℝn×n, with the diagonal entries equal to a ∈ ℝ and the off-diagonal entries equal to b ≥ 0. We provide shorter proofs for all the results therein using Fourier analysis. The key observation is that a circulant matrix is diagonalized by a DFT matrix. The results comprise an exact expression for ǁAǁp, 1 ≤ p ≤ ∞, where A = A(n, a, b), a ≥ 0 and for ǁAǁ2 where A = A(n, −a, b), a ≥ 0; for the other p-norms of A(n, −a, b), 2 < p < ∞, upper and lower bounds are derived.
{"title":"The p-norm of circulant matrices via Fourier analysis","authors":"K. R. Sahasranand","doi":"10.1515/conop-2021-0123","DOIUrl":"https://doi.org/10.1515/conop-2021-0123","url":null,"abstract":"Abstract A recent work derived expressions for the induced p-norm of a special class of circulant matrices A(n, a, b) ∈ ℝn×n, with the diagonal entries equal to a ∈ ℝ and the off-diagonal entries equal to b ≥ 0. We provide shorter proofs for all the results therein using Fourier analysis. The key observation is that a circulant matrix is diagonalized by a DFT matrix. The results comprise an exact expression for ǁAǁp, 1 ≤ p ≤ ∞, where A = A(n, a, b), a ≥ 0 and for ǁAǁ2 where A = A(n, −a, b), a ≥ 0; for the other p-norms of A(n, −a, b), 2 < p < ∞, upper and lower bounds are derived.","PeriodicalId":53800,"journal":{"name":"Concrete Operators","volume":"9 1","pages":"1 - 5"},"PeriodicalIF":0.6,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42749763","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 continuity of the embedding operator ℓ : ℋp(E) ↪ ℋ q(E) when 0 < p < q ⩽ ∞. The necessary and sufficient condition has already been described in [10] if p > 1. In this work, we address the problem when p = 1, using a new approach, but asking some additional hypothesis about the Hermite-Biehler function E. We give also a different proof for the case p > 1.
摘要研究了当0 < p < q≤∞时，嵌入算子h: h p(E)“h q(E)”的连续性。在[10]中已经描述了其充要条件。在这项工作中，我们使用一种新的方法解决了p = 1时的问题，但提出了一些关于Hermite-Biehler函数e的额外假设。
{"title":"Continuous embedding between P-de Branges spaces","authors":"Carlo Bellavita","doi":"10.1515/conop-2020-0118","DOIUrl":"https://doi.org/10.1515/conop-2020-0118","url":null,"abstract":"Abstract In this paper we study the continuity of the embedding operator ℓ : ℋp(E) ↪ ℋ q(E) when 0 < p < q ⩽ ∞. The necessary and sufficient condition has already been described in [10] if p > 1. In this work, we address the problem when p = 1, using a new approach, but asking some additional hypothesis about the Hermite-Biehler function E. We give also a different proof for the case p > 1.","PeriodicalId":53800,"journal":{"name":"Concrete Operators","volume":"8 1","pages":"125 - 134"},"PeriodicalIF":0.6,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44571930","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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：