We study linear and semi-linear wave, heat, and Schr"odinger equations defined by Kreu{i}n-Feller operator $-Delta_mu$ on a complete Riemannian $n$-manifolds $M$, where $mu$ is a finite positive Borel measure on a bounded open subset $Omega$ of $M$ with support contained in $overline{Omega}$. Under the assumption that $underline{operatorname{dim}}_{infty}(mu)>n-2$, we prove that for a linear or semi-linear equation of each of the above three types, there exists a unique weak solution. We study the crucial condition $dim_(mu)>n-2$ and provide examples of measures on $mathbb{S}^2$ and $mathbb{T}^2$ that satisfy the condition. We also study weak solutions of linear equations of the above three classes by using examples on $mathbb{S}^1$
{"title":"Differential equations defined by Kreĭn-Feller operators on Riemannian manifolds","authors":"Sze-Man Ngai, Lei Ouyang","doi":"arxiv-2408.04858","DOIUrl":"https://doi.org/arxiv-2408.04858","url":null,"abstract":"We study linear and semi-linear wave, heat, and Schr\"odinger equations\u0000defined by Kreu{i}n-Feller operator $-Delta_mu$ on a complete Riemannian\u0000$n$-manifolds $M$, where $mu$ is a finite positive Borel measure on a bounded\u0000open subset $Omega$ of $M$ with support contained in $overline{Omega}$.\u0000Under the assumption that $underline{operatorname{dim}}_{infty}(mu)>n-2$,\u0000we prove that for a linear or semi-linear equation of each of the above three\u0000types, there exists a unique weak solution. We study the crucial condition\u0000$dim_(mu)>n-2$ and provide examples of measures on $mathbb{S}^2$ and\u0000$mathbb{T}^2$ that satisfy the condition. We also study weak solutions of\u0000linear equations of the above three classes by using examples on $mathbb{S}^1$","PeriodicalId":501036,"journal":{"name":"arXiv - MATH - Functional Analysis","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141930537","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}
We study integral operators on the space of square-integrable functions from a compact set, $X$, to a separable Hilbert space, $H$. The kernel of such an operator takes values in the ideal of Hilbert-Schmidt operators on $H$. We establish regularity conditions on the kernel under which the associated integral operator is trace class. First, we extend Mercer's theorem to operator-valued kernels by proving that a continuous, nonnegative-definite, Hermitian symmetric kernel defines a trace class integral operator on $L^2(X;H)$ under an additional assumption. Second, we show that a general operator-valued kernel that is defined on a compact set and that is H"older continuous with H"older exponent greater than a half is trace class provided that the operator-valued kernel is essentially bounded as a mapping into the space of trace class operators on $H$. Finally, when $dim H < infty$, we show that an analogous result also holds for matrix-valued kernels on the real line, provided that an additional exponential decay assumption holds.
我们研究从紧凑集$X$到可分离希尔伯特空间$H$的平方可积分函数空间上的积分算子。这种算子的核在$H$上的希尔伯特-施密特算子理想中取值。我们建立了核的正则性条件，在此条件下，相关的积分算子是迹类的。首先，我们通过证明连续、非负有限、赫米特对称核在附加假设下定义了$L^2(X;H)$上的迹类积分算子，将默瑟定理扩展到了有算子值的核。其次，我们证明了一个定义在紧凑集上的一般算子值核是痕量类的，它是（H）连续的，且（H）指数大于一半，条件是算子值核作为映射到$H$上痕量类算子空间的映射本质上是有界的。最后，当$dim H < infty$时，我们证明了一个类似的结果也适用于实线上的矩阵值核，条件是一个额外的指数衰减假设成立。
{"title":"A regularity condition under which integral operators with operator-valued kernels are trace class","authors":"John Zweck, Yuri Latushkin, Erika Gallo","doi":"arxiv-2408.04794","DOIUrl":"https://doi.org/arxiv-2408.04794","url":null,"abstract":"We study integral operators on the space of square-integrable functions from\u0000a compact set, $X$, to a separable Hilbert space, $H$. The kernel of such an\u0000operator takes values in the ideal of Hilbert-Schmidt operators on $H$. We\u0000establish regularity conditions on the kernel under which the associated\u0000integral operator is trace class. First, we extend Mercer's theorem to\u0000operator-valued kernels by proving that a continuous, nonnegative-definite,\u0000Hermitian symmetric kernel defines a trace class integral operator on\u0000$L^2(X;H)$ under an additional assumption. Second, we show that a general\u0000operator-valued kernel that is defined on a compact set and that is H\"older\u0000continuous with H\"older exponent greater than a half is trace class provided\u0000that the operator-valued kernel is essentially bounded as a mapping into the\u0000space of trace class operators on $H$. Finally, when $dim H < infty$, we show\u0000that an analogous result also holds for matrix-valued kernels on the real line,\u0000provided that an additional exponential decay assumption holds.","PeriodicalId":501036,"journal":{"name":"arXiv - MATH - Functional Analysis","volume":"26 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141930535","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}
This is an expository-survey on weak stability of bounded linear operators acting on normed spaces in general and, in particular, on Hilbert spaces. The paper gives a comprehensive account of the problem of weak operator stability, containing a few new results and some unanswered questions. It also gives an updated review of the literature on the weak stability of operators over the past sixty years, including present-day research trends. It is verified that the majority of the weak stability literature is concentrated on Hilbert-space operators. We discuss why this preference occurs and also why the weak stability of unitary operators is central to the Hilbert-space stability problem.
{"title":"An Exposition on Weak Stability of Operators","authors":"C. S. Kubrusly","doi":"arxiv-2408.04186","DOIUrl":"https://doi.org/arxiv-2408.04186","url":null,"abstract":"This is an expository-survey on weak stability of bounded linear operators\u0000acting on normed spaces in general and, in particular, on Hilbert spaces. The\u0000paper gives a comprehensive account of the problem of weak operator stability,\u0000containing a few new results and some unanswered questions. It also gives an\u0000updated review of the literature on the weak stability of operators over the\u0000past sixty years, including present-day research trends. It is verified that\u0000the majority of the weak stability literature is concentrated on Hilbert-space\u0000operators. We discuss why this preference occurs and also why the weak\u0000stability of unitary operators is central to the Hilbert-space stability\u0000problem.","PeriodicalId":501036,"journal":{"name":"arXiv - MATH - Functional Analysis","volume":"56 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141930538","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}
We prove new monotonicity properties for spectral radius, essential spectral radius, operator norm, Hausdorff measure of non-compactness and numerical radius of products and sums of weighted geometric symmetrizations of positive kernel operators on $L^2$. To our knowledge, several proved properties are new even in the finite dimensional case.
{"title":"Monotonicity properties of weighted geometric symmetrizations","authors":"Katarina Bogdanović, Aljoša Peperko","doi":"arxiv-2408.04357","DOIUrl":"https://doi.org/arxiv-2408.04357","url":null,"abstract":"We prove new monotonicity properties for spectral radius, essential spectral\u0000radius, operator norm, Hausdorff measure of non-compactness and numerical\u0000radius of products and sums of weighted geometric symmetrizations of positive\u0000kernel operators on $L^2$. To our knowledge, several proved properties are new\u0000even in the finite dimensional case.","PeriodicalId":501036,"journal":{"name":"arXiv - MATH - Functional Analysis","volume":"368 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141930632","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}
This is a paper in a series that studies smooth relative Lie algebra homologies and cohomologies based on the theory of formal manifolds and formal Lie groups. In two previous papers, we develop the basic theory of formal manifolds, including generalizations of vector-valued distributions and generalized functions on smooth manifolds to the setting of formal manifolds. In this paper, we establish Poincar'e's lemma for de Rham complexes with coefficients in formal functions, formal generalized functions, compactly supported formal densities, or compactly supported formal distributions.
本文是基于形式流形和形式李群理论研究光滑相对李代数同调与同调的系列论文之一。在前两篇论文中，我们发展了形式流形的基本理论，包括将光滑流形上的向量值分布和广义函数推广到形式流形的环境中。在本文中，我们建立了以形式函数、形式广义函数、紧凑支持的形式密度或紧凑支持的形式分布为系数的 de Rham 复数的 Poincar'e' Lemma。
{"title":"Poincaré's lemma for formal manifolds","authors":"Fulin Chen, Binyong Sun, Chuyun Wang","doi":"arxiv-2408.04263","DOIUrl":"https://doi.org/arxiv-2408.04263","url":null,"abstract":"This is a paper in a series that studies smooth relative Lie algebra\u0000homologies and cohomologies based on the theory of formal manifolds and formal\u0000Lie groups. In two previous papers, we develop the basic theory of formal\u0000manifolds, including generalizations of vector-valued distributions and\u0000generalized functions on smooth manifolds to the setting of formal manifolds.\u0000In this paper, we establish Poincar'e's lemma for de Rham complexes with\u0000coefficients in formal functions, formal generalized functions, compactly\u0000supported formal densities, or compactly supported formal distributions.","PeriodicalId":501036,"journal":{"name":"arXiv - MATH - Functional Analysis","volume":"25 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141930536","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}
Let $sigma : mathbb C^d rightarrow mathbb C^d$ be an affine-linear involution such that $J_sigma = -1$ and let $U, V$ be two domains in $mathbb C^d$ with $U$ being $sigma$-invariant. Let $phi : U rightarrow V$ be a $sigma$-invariant $2$-proper map such that $J_phi$ is affine-linear and let $mathscr H(U)$ be a $sigma$-invariant reproducing kernel Hilbert space of complex-valued holomorphic functions on $U.$ It is shown that the space $mathscr H_phi(V):={f in mathrm{Hol}(V) : J_phi cdot f circ phi in mathscr H(U)}$ endowed with the norm $|f|_phi :=|J_phi cdot f circ phi|_{mathscr H(U)}$ is a reproducing kernel Hilbert space and the linear mapping $varGamma_phi$ defined by $ varGamma_phi(f) = J_phi cdot f circ phi,$ $f in mathrm{Hol}(V),$ is a unitary from $mathscr H_phi(V)$ onto ${f in mathscr H(U) : f = -f circ sigma}.$ Moreover, a neat formula for the reproducing kernel $kappa_{phi}$ of $mathscr H_phi(V)$ in terms of the reproducing kernel of $mathscr H(U)$ is given. The above scheme is applicable to symmetrized bidisc, tetrablock, $d$-dimensional fat Hartogs triangle and $d$-dimensional egg domain. This recovers some known results. Our result not only yields a candidate for Hardy spaces but also an analog of von Neumann's inequality for contractive tuples naturally associated with these domains. Unlike the existing techniques, we capitalize on the methods from several complex variables.
{"title":"A transference principle for involution-invariant functional Hilbert spaces","authors":"Santu Bera, Sameer Chavan, Shubham Jain","doi":"arxiv-2408.04384","DOIUrl":"https://doi.org/arxiv-2408.04384","url":null,"abstract":"Let $sigma : mathbb C^d rightarrow mathbb C^d$ be an affine-linear\u0000involution such that $J_sigma = -1$ and let $U, V$ be two domains in $mathbb\u0000C^d$ with $U$ being $sigma$-invariant. Let $phi : U rightarrow V$ be a\u0000$sigma$-invariant $2$-proper map such that $J_phi$ is affine-linear and let\u0000$mathscr H(U)$ be a $sigma$-invariant reproducing kernel Hilbert space of\u0000complex-valued holomorphic functions on $U.$ It is shown that the space\u0000$mathscr H_phi(V):={f in mathrm{Hol}(V) : J_phi cdot f circ phi in\u0000mathscr H(U)}$ endowed with the norm $|f|_phi :=|J_phi cdot f circ\u0000phi|_{mathscr H(U)}$ is a reproducing kernel Hilbert space and the linear\u0000mapping $varGamma_phi$ defined by $ varGamma_phi(f) = J_phi cdot f circ\u0000phi,$ $f in mathrm{Hol}(V),$ is a unitary from $mathscr H_phi(V)$ onto\u0000${f in mathscr H(U) : f = -f circ sigma}.$ Moreover, a neat formula for\u0000the reproducing kernel $kappa_{phi}$ of $mathscr H_phi(V)$ in terms of the\u0000reproducing kernel of $mathscr H(U)$ is given. The above scheme is applicable\u0000to symmetrized bidisc, tetrablock, $d$-dimensional fat Hartogs triangle and\u0000$d$-dimensional egg domain. This recovers some known results. Our result not\u0000only yields a candidate for Hardy spaces but also an analog of von Neumann's\u0000inequality for contractive tuples naturally associated with these domains.\u0000Unlike the existing techniques, we capitalize on the methods from several\u0000complex variables.","PeriodicalId":501036,"journal":{"name":"arXiv - MATH - Functional Analysis","volume":"10 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141930539","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}
Based on the rapid development of dyadic analysis and the theory of variable weighted function spaces over the spaces of homogeneous type $(X,d,mu)$ in recent years, we systematically consider the quantitative variable weighted characterizations for fractional maximal operators. On the one hand, a new class of variable multiple weight $A_{vec{p}(cdot),q(cdot)}(X)$ is established, which enables us to prove the strong and weak type variable multiple weighted estimates for multilinear fractional maximal operators ${{{mathscr M}_{eta }}}$. More precisely, [ {left[ {vec omega } right]_{{A_{vec p( cdot ),q( cdot )}}(X)}} lesssim {left| mathscr{M}_eta right|_{prodlimits_{i = 1}^m {{L^{p_i( cdot )}}({X,omega _i})} to {L^{q( cdot )}}(X,omega )({WL^{q( cdot )}}(X,omega ))}} le {C_{vec omega ,eta ,m,mu ,X,vec p( cdot )}}. ] On the other hand, on account of the classical Sawyer's condition $S_{p,q}(mathbb{R}^n)$, a new variable testing condition $C_{{p}(cdot),q(cdot)}(X)$ also appears in here, which allows us to obtain quantitative two-weighted estimates for fractional maximal operators ${{{M}_{eta }}}$. To be exact, begin{align*} |M_{eta}|_{L^{p(cdot)}(X,omega)rightarrow L^{q(cdot)}(X,v)} lesssim sum_{theta=frac{1}{{{p_{rm{ - }}}}},frac{1}{{{p_{rm{ + }}}}}} left([omega ]_{C_{p( cdot ),q( cdot )}^1(X)} [omega, v]_{C_{p(cdot), q(cdot)}^2(X)}right)^{theta}, end{align*} The implicit constants mentioned above are independent on the weights.
{"title":"New variable weighted conditions for fractional maximal operators over spaces of homogeneous type","authors":"Xi Cen","doi":"arxiv-2408.04544","DOIUrl":"https://doi.org/arxiv-2408.04544","url":null,"abstract":"Based on the rapid development of dyadic analysis and the theory of variable\u0000weighted function spaces over the spaces of homogeneous type $(X,d,mu)$ in\u0000recent years, we systematically consider the quantitative variable weighted\u0000characterizations for fractional maximal operators. On the one hand, a new\u0000class of variable multiple weight $A_{vec{p}(cdot),q(cdot)}(X)$ is\u0000established, which enables us to prove the strong and weak type variable\u0000multiple weighted estimates for multilinear fractional maximal operators\u0000${{{mathscr M}_{eta }}}$. More precisely, [ {left[ {vec omega }\u0000right]_{{A_{vec p( cdot ),q( cdot )}}(X)}} lesssim {left|\u0000mathscr{M}_eta right|_{prodlimits_{i = 1}^m {{L^{p_i( cdot )}}({X,omega\u0000_i})} to {L^{q( cdot )}}(X,omega )({WL^{q( cdot )}}(X,omega ))}} le\u0000{C_{vec omega ,eta ,m,mu ,X,vec p( cdot )}}. ] On the other hand, on account of the classical Sawyer's condition\u0000$S_{p,q}(mathbb{R}^n)$, a new variable testing condition\u0000$C_{{p}(cdot),q(cdot)}(X)$ also appears in here, which allows us to obtain\u0000quantitative two-weighted estimates for fractional maximal operators\u0000${{{M}_{eta }}}$. To be exact, begin{align*} |M_{eta}|_{L^{p(cdot)}(X,omega)rightarrow L^{q(cdot)}(X,v)} lesssim\u0000sum_{theta=frac{1}{{{p_{rm{ - }}}}},frac{1}{{{p_{rm{ + }}}}}}\u0000left([omega ]_{C_{p( cdot ),q( cdot )}^1(X)} [omega, v]_{C_{p(cdot),\u0000q(cdot)}^2(X)}right)^{theta}, end{align*} The implicit constants mentioned\u0000above are independent on the weights.","PeriodicalId":501036,"journal":{"name":"arXiv - MATH - Functional Analysis","volume":"14 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141949475","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}
Collective versions of order convergences and corresponding types of collectively qualified sets of operators in vector lattices are investigated. It is proved that every collectively order continuous set of operators between Archimedean vector lattices is collectively order bounded.
证明了阿基米德向量网格之间的每一个运算符集合阶连续集都是阶有界的。
{"title":"Collective order convergence and collectively qualified set of operators","authors":"Eduard Emelyanov","doi":"arxiv-2408.03671","DOIUrl":"https://doi.org/arxiv-2408.03671","url":null,"abstract":"Collective versions of order convergences and corresponding types of\u0000collectively qualified sets of operators in vector lattices are investigated.\u0000It is proved that every collectively order continuous set of operators between\u0000Archimedean vector lattices is collectively order bounded.","PeriodicalId":501036,"journal":{"name":"arXiv - MATH - Functional Analysis","volume":"84 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141930556","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}
In this paper we derive families of Gershgorin-type inclusion sets for the spectra and pseudospectra of finite matrices. In common with previous generalisations of the classical Gershgorin bound for the spectrum, our inclusion sets are based on a block decomposition. In contrast to previous generalisations that treat the matrix as a perturbation of a block-diagonal submatrix, our arguments treat the matrix as a perturbation of a block-tridiagonal matrix, which can lead to sharp spectral bounds, as we show for the example of large Toeplitz matrices. Our inclusion sets, which take the form of unions of pseudospectra of square or rectangular submatrices, build on our own recent work on inclusion sets for bi-infinite matrices [Chandler-Wilde, Chonchaiya, Lindner, {em J. Spectr. Theory} {bf 14}, 719--804 (2024)].
在本文中，我们推导出了有限矩阵谱和伪谱的格什高林型包含集系列。与之前对谱的经典格什高林约束的概括一样，我们的包含集基于块分解。与之前将矩阵视为块对角线子矩阵扰动的概括不同，我们的论证将矩阵视为块对角线矩阵的扰动，这可以导致尖锐的谱约束，正如我们以大型托普利兹矩阵为例所展示的那样。我们的包含集是正方形或矩形子矩阵伪谱的联合形式，建立在我们自己最近关于双无限矩阵包含集的工作之上[Chandler-Wilde, Chonchaiya, Lindner, {em J. Spectr.Theory}{bf 14}, 719--804 (2024)].
{"title":"Gershgorin-Type Spectral Inclusions for Matrices","authors":"Simon N. Chandler-Wilde, Marko Lindner","doi":"arxiv-2408.03883","DOIUrl":"https://doi.org/arxiv-2408.03883","url":null,"abstract":"In this paper we derive families of Gershgorin-type inclusion sets for the\u0000spectra and pseudospectra of finite matrices. In common with previous\u0000generalisations of the classical Gershgorin bound for the spectrum, our\u0000inclusion sets are based on a block decomposition. In contrast to previous\u0000generalisations that treat the matrix as a perturbation of a block-diagonal\u0000submatrix, our arguments treat the matrix as a perturbation of a\u0000block-tridiagonal matrix, which can lead to sharp spectral bounds, as we show\u0000for the example of large Toeplitz matrices. Our inclusion sets, which take the\u0000form of unions of pseudospectra of square or rectangular submatrices, build on\u0000our own recent work on inclusion sets for bi-infinite matrices [Chandler-Wilde,\u0000Chonchaiya, Lindner, {em J. Spectr. Theory} {bf 14}, 719--804 (2024)].","PeriodicalId":501036,"journal":{"name":"arXiv - MATH - Functional Analysis","volume":"32 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141930553","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}
Let M"ob be the biholomorphic automorphism group of the unit disc of the complex plane, $mathcal{H}$ be a complex separable Hilbert space and $mathcal{U}(mathcal{H})$ be the group of all unitary operators. Suppose $mathcal{H}$ is a reproducing kernel Hilbert space consisting of holomorphic functions over the poly-disc $mathbb D^n$ and contains all the polynomials. If $pi : mbox{M"ob} to mathcal{U}(mathcal{H})$ is a multiplier representation, then we prove that there exist $lambda_1, lambda_2, ldots, lambda_n > 0$ such that $pi$ is unitarily equivalent to $(otimes_{i=1}^{n} D_{lambda_i}^+)|_{mbox{M"ob}}$, where each $D_{lambda_i}^+$ is a holomorphic discrete series representation of M"ob. As an application, we prove that if $(T_1, T_2)$ is a M"ob - homogeneous pair in the Cowen - Douglas class of rank $1$ over the bi-disc, then each $T_i$ posses an upper triangular form with respect to a decomposition of the Hilbert space. In this upper triangular form of each $T_i$, the diagonal operators are identified. We also prove that if $mathcal{H}$ consists of symmetric (resp. anti-symmetric) holomorphic functions over $mathbb D^2$ and contains all the symmetric (resp. anti-symmetric) polynomials, then there exists $lambda > 0$ such that $pi cong oplus_{m = 0}^infty D^+_{lambda + 4m}$ (resp. $pi cong oplus_{m=0}^infty D^+_{lambda + 4m + 2}$).
{"title":"Representations of the Möbius group and pairs of homogeneous operators in the Cowen-Douglas class","authors":"Jyotirmay Das, Somnath Hazra","doi":"arxiv-2408.03711","DOIUrl":"https://doi.org/arxiv-2408.03711","url":null,"abstract":"Let M\"ob be the biholomorphic automorphism group of the unit disc of the\u0000complex plane, $mathcal{H}$ be a complex separable Hilbert space and\u0000$mathcal{U}(mathcal{H})$ be the group of all unitary operators. Suppose\u0000$mathcal{H}$ is a reproducing kernel Hilbert space consisting of holomorphic\u0000functions over the poly-disc $mathbb D^n$ and contains all the polynomials. If\u0000$pi : mbox{M\"ob} to mathcal{U}(mathcal{H})$ is a multiplier\u0000representation, then we prove that there exist $lambda_1, lambda_2, ldots,\u0000lambda_n > 0$ such that $pi$ is unitarily equivalent to $(otimes_{i=1}^{n}\u0000D_{lambda_i}^+)|_{mbox{M\"ob}}$, where each $D_{lambda_i}^+$ is a\u0000holomorphic discrete series representation of M\"ob. As an application, we\u0000prove that if $(T_1, T_2)$ is a M\"ob - homogeneous pair in the Cowen - Douglas\u0000class of rank $1$ over the bi-disc, then each $T_i$ posses an upper triangular\u0000form with respect to a decomposition of the Hilbert space. In this upper\u0000triangular form of each $T_i$, the diagonal operators are identified. We also\u0000prove that if $mathcal{H}$ consists of symmetric (resp. anti-symmetric)\u0000holomorphic functions over $mathbb D^2$ and contains all the symmetric (resp.\u0000anti-symmetric) polynomials, then there exists $lambda > 0$ such that $pi\u0000cong oplus_{m = 0}^infty D^+_{lambda + 4m}$ (resp. $pi cong\u0000oplus_{m=0}^infty D^+_{lambda + 4m + 2}$).","PeriodicalId":501036,"journal":{"name":"arXiv - MATH - Functional Analysis","volume":"75 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141930555","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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