Abstract We introduce a concept of hereditary set of multi-indices, and consider vector spaces of functions generated by families associated to such sets of multi-indices, called hereditary function spaces. Existence and uniquenes of representing measures for some abstract truncated moment problems are investigated in this framework, by adapting the concept of idempotent and that of dimensional stability, and using some techniques involving C*-algebras and commuting self-adjoint multiplication operators.
{"title":"Moment Problems in Hereditary Function Spaces","authors":"F. Vasilescu","doi":"10.1515/conop-2019-0006","DOIUrl":"https://doi.org/10.1515/conop-2019-0006","url":null,"abstract":"Abstract We introduce a concept of hereditary set of multi-indices, and consider vector spaces of functions generated by families associated to such sets of multi-indices, called hereditary function spaces. Existence and uniquenes of representing measures for some abstract truncated moment problems are investigated in this framework, by adapting the concept of idempotent and that of dimensional stability, and using some techniques involving C*-algebras and commuting self-adjoint multiplication operators.","PeriodicalId":53800,"journal":{"name":"Concrete Operators","volume":"6 1","pages":"64 - 75"},"PeriodicalIF":0.6,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/conop-2019-0006","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49030342","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 situations, when two unbounded operators generated by infinite Jacobi matrices, are self-adjoint and commute. It is found that if two Jacobi matrices formally commute, then two corresponding operators are either self-adjoint and commute, or admit a commuting self-adjoint extensions. In the latter case such extensions are explicitly described. Also, some necessary and sufficient conditions for self-adjointness of Jacobi operators are studied.
{"title":"On unbounded commuting Jacobi operators and some related issues","authors":"A. Osipov","doi":"10.1515/conop-2019-0008","DOIUrl":"https://doi.org/10.1515/conop-2019-0008","url":null,"abstract":"Abstract We consider the situations, when two unbounded operators generated by infinite Jacobi matrices, are self-adjoint and commute. It is found that if two Jacobi matrices formally commute, then two corresponding operators are either self-adjoint and commute, or admit a commuting self-adjoint extensions. In the latter case such extensions are explicitly described. Also, some necessary and sufficient conditions for self-adjointness of Jacobi operators are studied.","PeriodicalId":53800,"journal":{"name":"Concrete Operators","volume":"6 1","pages":"82 - 91"},"PeriodicalIF":0.6,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/conop-2019-0008","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44236007","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 Berezin transform à of an operator A, acting on the reproducing kernel Hilbert space ℋ = ℋ (Ω) over some (non-empty) set Ω, is defined by Ã(λ) = 〉Aǩ λ, ǩ λ〈 (λ ∈ Ω), where k⌢λ=kλ‖ kλ ‖ ${mathord{buildrel{lower3pthbox{$scriptscriptstylefrown$}}over k} _lambda } = {{{k_lambda }} over {left| {{k_lambda }} right|}}$ is the normalized reproducing kernel of ℋ. The Berezin number of an operator A is defined by ber(A)=supλ∈Ω| A˜(λ) |=supλ∈Ω| 〈 Ak⌢λ,k⌢λ 〉 | ${bf{ber}}{rm{(}}A) = mathop {sup }limits_{lambda in Omega } left| {tilde A(lambda )} right| = mathop {sup }limits_{lambda in Omega } left| {leftlangle {A{{mathord{buildrel{lower3pthbox{$scriptscriptstylefrown$}}over k} }_lambda },{{mathord{buildrel{lower3pthbox{$scriptscriptstylefrown$}}over k} }_lambda }} rightrangle } right|$ . In this paper, we prove some Berezin number inequalities. Among other inequalities, it is shown that if A, B, X are bounded linear operators on a Hilbert space ℋ, then ber(AX±XA)⩽ber12(A*A+AA*)ber12(X*X+XX*) $${bf{ber}}(AX pm XA) leqslant {bf{be}}{{bf{r}}^{{1 over 2}}}left( {A*A + AA*} right){bf{be}}{{bf{r}}^{{1 over 2}}}left( {X*X + XX*} right)$$ and ber2(A*XB)⩽‖ X ‖2ber(A*A)ber(B*B). $${bf{be}}{{bf{r}}^2}({A^*}XB) leqslant {left| X right|^2}{bf{ber}}({A^*}A){bf{ber}}({B^*}B).$$ We also prove the multiplicative inequality ber(AB)⩽ber(A)ber(B) $${bf{ber}}(AB){bf{ber}}(A){bf{ber}}(B)$$
算子A的Berezin变换Ã作用于某个(非空)集Ω上的再现核希尔伯特空间h = h (Ω)，定义为Ã(λ) = > a λ， λ < (λ∈Ω)，其中kλ =kλ‖kλ‖ ${mathord{buildrel{lower3pthbox{$scriptscriptstylefrown$}}over k} _lambda } = {{{k_lambda }} over {left| {{k_lambda }} right|}}$ 是h的归一化再现核。算子A的Berezin数定义为ber(A)=supλ∈Ω| A ～ (λ) |=supλ∈Ω| < Ak λ，k λ > | ${bf{ber}}{rm{(}}A) = mathop {sup }limits_{lambda in Omega } left| {tilde A(lambda )} right| = mathop {sup }limits_{lambda in Omega } left| {leftlangle {A{{mathord{buildrel{lower3pthbox{$scriptscriptstylefrown$}}over k} }_lambda },{{mathord{buildrel{lower3pthbox{$scriptscriptstylefrown$}}over k} }_lambda }} rightrangle } right|$ 。本文证明了一些Berezin数不等式。在其他不等式中，证明了如果A, B, X是Hilbert空间h上的有界线性算子，则ber(AX±XA)≤ber12(A*A+AA*)ber12(X*X+XX*) $${bf{ber}}(AX pm XA) leqslant {bf{be}}{{bf{r}}^{{1 over 2}}}left( {A*A + AA*} right){bf{be}}{{bf{r}}^{{1 over 2}}}left( {X*X + XX*} right)$$ ber2(A*XB)≥‖X‖2ber(A*A)ber(B*B)。 $${bf{be}}{{bf{r}}^2}({A^*}XB) leqslant {left| X right|^2}{bf{ber}}({A^*}A){bf{ber}}({B^*}B).$$ 我们还证明了乘法不等式ber(AB)≤ber(A)ber(B) $${bf{ber}}(AB){bf{ber}}(A){bf{ber}}(B)$$
{"title":"Berezin number inequalities for operators","authors":"M. Bakherad, M. Garayev","doi":"10.1515/conop-2019-0003","DOIUrl":"https://doi.org/10.1515/conop-2019-0003","url":null,"abstract":"Abstract The Berezin transform à of an operator A, acting on the reproducing kernel Hilbert space ℋ = ℋ (Ω) over some (non-empty) set Ω, is defined by Ã(λ) = 〉Aǩ λ, ǩ λ〈 (λ ∈ Ω), where k⌢λ=kλ‖ kλ ‖ ${mathord{buildrel{lower3pthbox{$scriptscriptstylefrown$}}over k} _lambda } = {{{k_lambda }} over {left| {{k_lambda }} right|}}$ is the normalized reproducing kernel of ℋ. The Berezin number of an operator A is defined by ber(A)=supλ∈Ω| A˜(λ) |=supλ∈Ω| 〈 Ak⌢λ,k⌢λ 〉 | ${bf{ber}}{rm{(}}A) = mathop {sup }limits_{lambda in Omega } left| {tilde A(lambda )} right| = mathop {sup }limits_{lambda in Omega } left| {leftlangle {A{{mathord{buildrel{lower3pthbox{$scriptscriptstylefrown$}}over k} }_lambda },{{mathord{buildrel{lower3pthbox{$scriptscriptstylefrown$}}over k} }_lambda }} rightrangle } right|$ . In this paper, we prove some Berezin number inequalities. Among other inequalities, it is shown that if A, B, X are bounded linear operators on a Hilbert space ℋ, then ber(AX±XA)⩽ber12(A*A+AA*)ber12(X*X+XX*) $${bf{ber}}(AX pm XA) leqslant {bf{be}}{{bf{r}}^{{1 over 2}}}left( {A*A + AA*} right){bf{be}}{{bf{r}}^{{1 over 2}}}left( {X*X + XX*} right)$$ and ber2(A*XB)⩽‖ X ‖2ber(A*A)ber(B*B). $${bf{be}}{{bf{r}}^2}({A^*}XB) leqslant {left| X right|^2}{bf{ber}}({A^*}A){bf{ber}}({B^*}B).$$ We also prove the multiplicative inequality ber(AB)⩽ber(A)ber(B) $${bf{ber}}(AB){bf{ber}}(A){bf{ber}}(B)$$","PeriodicalId":53800,"journal":{"name":"Concrete Operators","volume":"6 1","pages":"33 - 43"},"PeriodicalIF":0.6,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/conop-2019-0003","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42422999","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 (real or complex, separable) Banach space, and a contraction T on X, we say that T has the Blum-Hanson property if whenever x, y ∈ X are such that Tnx tends weakly to y in X as n tends to infinity, the means 1N∑k=1NTnkx {1 over N}sumlimits_{k = 1}^N {{T^{{n_k}}}x} tend to y in norm for every strictly increasing sequence (nk) k≥1 of integers. The space X itself has the Blum-Hanson property if every contraction on X has the Blum-Hanson property. We explain the ergodic-theoretic motivation for the Blum-Hanson property, prove that Hilbert spaces have the Blum-Hanson property, and then present a recent criterion of a geometric flavor, due to Lefèvre-Matheron-Primot, which allows to retrieve essentially all the known examples of spaces with the Blum-Hanson property. Lastly, following Lefèvre-Matheron, we characterize the compact metric spaces K such that the space C(K) has the Blum-Hanson property.
{"title":"The Blum-Hanson Property","authors":"S. Grivaux","doi":"10.1515/conop-2019-0009","DOIUrl":"https://doi.org/10.1515/conop-2019-0009","url":null,"abstract":"Abstract Given a (real or complex, separable) Banach space, and a contraction T on X, we say that T has the Blum-Hanson property if whenever x, y ∈ X are such that Tnx tends weakly to y in X as n tends to infinity, the means 1N∑k=1NTnkx {1 over N}sumlimits_{k = 1}^N {{T^{{n_k}}}x} tend to y in norm for every strictly increasing sequence (nk) k≥1 of integers. The space X itself has the Blum-Hanson property if every contraction on X has the Blum-Hanson property. We explain the ergodic-theoretic motivation for the Blum-Hanson property, prove that Hilbert spaces have the Blum-Hanson property, and then present a recent criterion of a geometric flavor, due to Lefèvre-Matheron-Primot, which allows to retrieve essentially all the known examples of spaces with the Blum-Hanson property. Lastly, following Lefèvre-Matheron, we characterize the compact metric spaces K such that the space C(K) has the Blum-Hanson property.","PeriodicalId":53800,"journal":{"name":"Concrete Operators","volume":"6 1","pages":"105 - 92"},"PeriodicalIF":0.6,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/conop-2019-0009","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45295023","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 present the existence of n-tuple of operators on complex Hilbert space that has a somewhere dense orbit and is not dense. We give the solution to the question stated in [11]: “Is there n-tuple of operators on a complex Hilbert space that has a somewhere dense orbit that is not dense?” We do so by extending the results due to Feldman [11] and Leòn-Saavedra [12] to complex Hilbert space. Further illustrative examples of somewhere dense orbits are given to support the results.
{"title":"Somewhere Dense Orbit that is not Dense on a Complex Hilbert Space","authors":"Neema Wilberth, Marco Mpimbo, Santosh Kumar","doi":"10.1515/conop-2019-0005","DOIUrl":"https://doi.org/10.1515/conop-2019-0005","url":null,"abstract":"Abstract In this paper, we present the existence of n-tuple of operators on complex Hilbert space that has a somewhere dense orbit and is not dense. We give the solution to the question stated in [11]: “Is there n-tuple of operators on a complex Hilbert space that has a somewhere dense orbit that is not dense?” We do so by extending the results due to Feldman [11] and Leòn-Saavedra [12] to complex Hilbert space. Further illustrative examples of somewhere dense orbits are given to support the results.","PeriodicalId":53800,"journal":{"name":"Concrete Operators","volume":"6 1","pages":"58 - 63"},"PeriodicalIF":0.6,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/conop-2019-0005","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49344022","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 concerns the analysis of the structure of bi-contractive projections on spaces of vector valued continuous functions and presents results that extend the characterization of bi-contractive projections given by the first author. It also includes a partial generalization of these results to affine and vector valued continuous functions from a Choquet simplex into a Hilbert space.
{"title":"A note on bi-contractive projections on spaces of vector valued continuous functions","authors":"F. Botelho, T. Rao","doi":"10.1515/conop-2018-0005","DOIUrl":"https://doi.org/10.1515/conop-2018-0005","url":null,"abstract":"Abstract This paper concerns the analysis of the structure of bi-contractive projections on spaces of vector valued continuous functions and presents results that extend the characterization of bi-contractive projections given by the first author. It also includes a partial generalization of these results to affine and vector valued continuous functions from a Choquet simplex into a Hilbert space.","PeriodicalId":53800,"journal":{"name":"Concrete Operators","volume":"5 1","pages":"42 - 49"},"PeriodicalIF":0.6,"publicationDate":"2018-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/conop-2018-0005","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44499822","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 N ( T ; V ) Nleft(T;hspace{0.33em}V) denote the number of eigenvalues of the Schrödinger operator − y ″ + V y -{y}^{^{primeprime} }+Vy with absolute value less than T T . This article studies the Weyl asymptotics of perturbations of the Schrödinger operator − y ″ + 1 4 e 2 t y -{y}^{^{primeprime} }+frac{1}{4}{e}^{2t}y on [ x 0 , ∞ ) left[{x}_{0},infty ) . In particular, we show that perturbations by functions ε ( t ) varepsilon left(t) that satisfy ∣ ε ( t ) ∣ ≲ e t | varepsilon left(t)| hspace{0.33em}lesssim hspace{0.33em}{e}^{t} do not change the Weyl asymptotics very much. Special emphasis is placed on connections to the asymptotics of the zeros of the Riemann zeta function.
{"title":"Weyl asymptotics for perturbations of Morse potential and connections to the Riemann zeta function","authors":"R. Rahm","doi":"10.1515/conop-2022-0139","DOIUrl":"https://doi.org/10.1515/conop-2022-0139","url":null,"abstract":"Abstract Let N ( T ; V ) Nleft(T;hspace{0.33em}V) denote the number of eigenvalues of the Schrödinger operator − y ″ + V y -{y}^{^{primeprime} }+Vy with absolute value less than T T . This article studies the Weyl asymptotics of perturbations of the Schrödinger operator − y ″ + 1 4 e 2 t y -{y}^{^{primeprime} }+frac{1}{4}{e}^{2t}y on [ x 0 , ∞ ) left[{x}_{0},infty ) . In particular, we show that perturbations by functions ε ( t ) varepsilon left(t) that satisfy ∣ ε ( t ) ∣ ≲ e t | varepsilon left(t)| hspace{0.33em}lesssim hspace{0.33em}{e}^{t} do not change the Weyl asymptotics very much. Special emphasis is placed on connections to the asymptotics of the zeros of the Riemann zeta function.","PeriodicalId":53800,"journal":{"name":"Concrete Operators","volume":"10 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2018-11-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46486561","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 Dirichlet problem on in_nite and locally _nite rooted trees, andwe prove that the set of irregular points for continuous data has zero capacity. We also give some uniqueness results for solutions in Sobolev W1,p of the tree.
{"title":"Some remarks on the Dirichlet problem on infinite trees","authors":"Nikolaos Chalmoukis, Matteo Levi","doi":"10.1515/conop-2019-0002","DOIUrl":"https://doi.org/10.1515/conop-2019-0002","url":null,"abstract":"Abstract We consider the Dirichlet problem on in_nite and locally _nite rooted trees, andwe prove that the set of irregular points for continuous data has zero capacity. We also give some uniqueness results for solutions in Sobolev W1,p of the tree.","PeriodicalId":53800,"journal":{"name":"Concrete Operators","volume":"6 1","pages":"20 - 32"},"PeriodicalIF":0.6,"publicationDate":"2018-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/conop-2019-0002","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46497583","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 explores the continuity and differentiability properties for the distribution function for a polynomial
摘要本文探讨了多项式分布函数的连续性和可微性
{"title":"The Distribution Function for a Polynomial","authors":"J. Cima, W. Derrick","doi":"10.1515/conop-2018-0004","DOIUrl":"https://doi.org/10.1515/conop-2018-0004","url":null,"abstract":"Abstract This paper explores the continuity and differentiability properties for the distribution function for a polynomial","PeriodicalId":53800,"journal":{"name":"Concrete Operators","volume":"5 1","pages":"35 - 41"},"PeriodicalIF":0.6,"publicationDate":"2018-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/conop-2018-0004","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49428908","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 properties of inner and outer functions in the Hardy space of the quaternionic unit ball. In particular, we give sufficient conditions as well as necessary ones for functions to be inner or outer.
{"title":"Quaternionic inner and outer functions","authors":"A. Monguzzi, G. Sarfatti, D. Seco","doi":"10.1515/conop-2019-0004","DOIUrl":"https://doi.org/10.1515/conop-2019-0004","url":null,"abstract":"Abstract We study properties of inner and outer functions in the Hardy space of the quaternionic unit ball. In particular, we give sufficient conditions as well as necessary ones for functions to be inner or outer.","PeriodicalId":53800,"journal":{"name":"Concrete Operators","volume":"6 1","pages":"44 - 57"},"PeriodicalIF":0.6,"publicationDate":"2018-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/conop-2019-0004","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45056067","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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