Selfadjoint and maximal dissipative extensions of a non-densely defined symmetric operator $S$ in an infinite-dimensional separable Hilbert space are considered and their compressions on the subspace ${rm overline{dom},} S$ are studied. The main focus is on the case ${rm codim,}{rm overline{dom},}S=infty$. New properties of the characteristic functions of non-densely defined symmetric operators are established.
{"title":"Compressions of selfadjoint and maximal dissipative extensions of non-densely defined symmetric operators","authors":"Yu. M. Arlinskiĭ","doi":"arxiv-2409.10234","DOIUrl":"https://doi.org/arxiv-2409.10234","url":null,"abstract":"Selfadjoint and maximal dissipative extensions of a non-densely defined\u0000symmetric operator $S$ in an infinite-dimensional separable Hilbert space are\u0000considered and their compressions on the subspace ${rm overline{dom},} S$\u0000are studied. The main focus is on the case ${rm codim,}{rm\u0000overline{dom},}S=infty$. New properties of the characteristic functions of\u0000non-densely defined symmetric operators are established.","PeriodicalId":501036,"journal":{"name":"arXiv - MATH - Functional Analysis","volume":"14 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142249329","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 explore the positive solutions of the following nonlinear Choquard equation involving the green kernel of the fractional operator $(-Delta_{mathbb{B}^N})^{-alpha/2}$ in the hyperbolic space begin{equation} begin{aligned} -Delta_{mathbb{B}^{N}} u , - , lambda u , &= left[(- Delta_{mathbb{B}^{N}})^{-frac{alpha}{2}}|u|^pright]|u|^{p-2}u, end{aligned} end{equation} where $Delta_{mathbb{B}^{N}}$ denotes the Laplace-Beltrami operator on $mathbb{B}^{N}$, $lambda leq frac{(N-1)^2}{4}$, $1 < p < 2^*_{alpha} = frac{N+alpha}{N-2}$, $0 < alpha < N$, $N geq 3$, $2^*_alpha$ is the critical exponent in the context of the Hardy-Littlewood-Sobolev inequality. This study is analogous to the Choquard equation in the Euclidean space, which involves the non-local Riesz potential operator. We consider the functional setting within the Sobolev space $H^1(mathbb{B}^N)$, employing advanced harmonic analysis techniques, particularly the Helgason Fourier transform and semigroup approach to fractional Laplacian. Moreover, the Hardy-Littlewood-Sobolev inequality on complete Riemannian manifolds, as developed by Varopoulos, is pivotal in our analysis. We prove an existence result for the above problem in the subcritical case. Moreover, we also demonstrate that solutions exhibit radial symmetry, and establish the regularity properties.
{"title":"Existence, symmetry and regularity of ground states of a non linear choquard equation in the hyperbolic space","authors":"Diksha Gupta, K. Sreenadh","doi":"arxiv-2409.10236","DOIUrl":"https://doi.org/arxiv-2409.10236","url":null,"abstract":"In this paper, we explore the positive solutions of the following nonlinear\u0000Choquard equation involving the green kernel of the fractional operator\u0000$(-Delta_{mathbb{B}^N})^{-alpha/2}$ in the hyperbolic space begin{equation} begin{aligned} -Delta_{mathbb{B}^{N}} u , - , lambda u , &= left[(-\u0000Delta_{mathbb{B}^{N}})^{-frac{alpha}{2}}|u|^pright]|u|^{p-2}u, end{aligned} end{equation} where $Delta_{mathbb{B}^{N}}$ denotes the Laplace-Beltrami\u0000operator on $mathbb{B}^{N}$, $lambda leq frac{(N-1)^2}{4}$, $1 < p <\u00002^*_{alpha} = frac{N+alpha}{N-2}$, $0 < alpha < N$, $N geq 3$,\u0000$2^*_alpha$ is the critical exponent in the context of the\u0000Hardy-Littlewood-Sobolev inequality. This study is analogous to the Choquard\u0000equation in the Euclidean space, which involves the non-local Riesz potential\u0000operator. We consider the functional setting within the Sobolev space\u0000$H^1(mathbb{B}^N)$, employing advanced harmonic analysis techniques,\u0000particularly the Helgason Fourier transform and semigroup approach to\u0000fractional Laplacian. Moreover, the Hardy-Littlewood-Sobolev inequality on\u0000complete Riemannian manifolds, as developed by Varopoulos, is pivotal in our\u0000analysis. We prove an existence result for the above problem in the subcritical\u0000case. Moreover, we also demonstrate that solutions exhibit radial symmetry, and\u0000establish the regularity properties.","PeriodicalId":501036,"journal":{"name":"arXiv - MATH - Functional Analysis","volume":"53 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142249338","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}
For the $N$-dimensional Fourier matrix $mathcal F_N$, we show that if $N geq 2$, then all $2times 2$ principal minors of $mathcal F_N$ are nonzero if and only if $N$ is square-free. Additionally, we show that if $N > 4$, then all $3times 3$ principal minors of $mathcal F_N$ are nonzero if and only if $N$ is square-free. Moreover, based on numerical experiments, we conjecture that if $N$ is square-free, then all principal minors of $mathcal F_N$ are nonzero.
{"title":"On the principal minors of Fourier matrices","authors":"Andrei Caragea, Dae Gwan Lee","doi":"arxiv-2409.09793","DOIUrl":"https://doi.org/arxiv-2409.09793","url":null,"abstract":"For the $N$-dimensional Fourier matrix $mathcal F_N$, we show that if $N\u0000geq 2$, then all $2times 2$ principal minors of $mathcal F_N$ are nonzero if\u0000and only if $N$ is square-free. Additionally, we show that if $N > 4$, then all\u0000$3times 3$ principal minors of $mathcal F_N$ are nonzero if and only if $N$\u0000is square-free. Moreover, based on numerical experiments, we conjecture that if\u0000$N$ is square-free, then all principal minors of $mathcal F_N$ are nonzero.","PeriodicalId":501036,"journal":{"name":"arXiv - MATH - Functional Analysis","volume":"2 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142249332","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}
Aminallah Khosravi, Hamid Reza Ebrahimi Vishki, Ramin Faal
We say that a Banach algebra A has $k$-orthogonally additive property ($k$-OA property, for short) if every orthogonally additive k-homogeneous polynomial $P:mathcal{A}to mathbb{C}$ can be expressed in the standard form $P(x)=langle gamma,x^krangle$, $(xin mathcal{A})$, for some $gammain mathcal{A}^*$. In this paper we first investigate the extensions of a $k$-homogeneous polynomial from $mathcal{A}$ to the bidual $mathcal{A}^{**}$; equipped with the first Arens product. We then study the relationship between $k$-OA properties of $mathcal{A}$ and $mathcal{A}^{**}$: This relation is specially investigated for a dual Banach algebra. Finally we examine our results for the dual Banach algebra $ell^{1}$, with pointwise product, and we show that the Banach algebra $(ell^{1})^{**}$ enjoys k-OA property.
{"title":"Orthogonally additive polynomials on the bidual of Banach algebras","authors":"Aminallah Khosravi, Hamid Reza Ebrahimi Vishki, Ramin Faal","doi":"arxiv-2409.09711","DOIUrl":"https://doi.org/arxiv-2409.09711","url":null,"abstract":"We say that a Banach algebra A has $k$-orthogonally additive property ($k$-OA\u0000property, for short) if every orthogonally additive k-homogeneous polynomial\u0000$P:mathcal{A}to mathbb{C}$ can be expressed in the standard form\u0000$P(x)=langle gamma,x^krangle$, $(xin mathcal{A})$, for some $gammain\u0000mathcal{A}^*$. In this paper we first investigate the extensions of a\u0000$k$-homogeneous polynomial from $mathcal{A}$ to the bidual $mathcal{A}^{**}$;\u0000equipped with the first Arens product. We then study the relationship between\u0000$k$-OA properties of $mathcal{A}$ and $mathcal{A}^{**}$: This relation is\u0000specially investigated for a dual Banach algebra. Finally we examine our\u0000results for the dual Banach algebra $ell^{1}$, with pointwise product, and we\u0000show that the Banach algebra $(ell^{1})^{**}$ enjoys k-OA property.","PeriodicalId":501036,"journal":{"name":"arXiv - MATH - Functional Analysis","volume":"2 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142249334","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 present a general and natural framework to study the dynamics of composition operators on spaces of measurable functions, in which we then reconsider the characterizations for hypercyclic and mixing composition operators obtained by Bayart, Darji and Pires. We show that the notions of hypercyclicity and weak mixing coincide in this context and, if the system is dissipative, the recurrent composition operators agree with the hypercyclic ones. We also give a characterization for invertible composition operators satisfying Kitai's Criterion, and we construct an example of a mixing composition operator not satisfying Kitai's Criterion. For invertible dissipative systems with bounded distortion we show that composition operators satisfying Kitai's Criterion coincide with the mixing operators.
{"title":"Kitai's Criterion for Composition Operators","authors":"Daniel Gomes, Karl-G. Grosse-Erdmann","doi":"arxiv-2409.09443","DOIUrl":"https://doi.org/arxiv-2409.09443","url":null,"abstract":"We present a general and natural framework to study the dynamics of\u0000composition operators on spaces of measurable functions, in which we then\u0000reconsider the characterizations for hypercyclic and mixing composition\u0000operators obtained by Bayart, Darji and Pires. We show that the notions of\u0000hypercyclicity and weak mixing coincide in this context and, if the system is\u0000dissipative, the recurrent composition operators agree with the hypercyclic\u0000ones. We also give a characterization for invertible composition operators\u0000satisfying Kitai's Criterion, and we construct an example of a mixing\u0000composition operator not satisfying Kitai's Criterion. For invertible\u0000dissipative systems with bounded distortion we show that composition operators\u0000satisfying Kitai's Criterion coincide with the mixing operators.","PeriodicalId":501036,"journal":{"name":"arXiv - MATH - Functional Analysis","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142249333","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}
Timothy G. Clos, Zeljko Cuckovic, Sonmez Sahutoglu
Let $varphi$ be a holomorphic self map of the bidisc that is Lipschitz on the closure. We show that the composition operator $C_{varphi}$ is compact on the Bergman space if and only if $varphi(overline{mathbb{D}^2})cap mathbb{T}^2=emptyset$ and $varphi(overline{mathbb{D}^2}setminus mathbb{T}^2)cap bmathbb{D}^2=emptyset$.
{"title":"Compactness of composition operators on the Bergman space of the bidisc","authors":"Timothy G. Clos, Zeljko Cuckovic, Sonmez Sahutoglu","doi":"arxiv-2409.09529","DOIUrl":"https://doi.org/arxiv-2409.09529","url":null,"abstract":"Let $varphi$ be a holomorphic self map of the bidisc that is Lipschitz on\u0000the closure. We show that the composition operator $C_{varphi}$ is compact on\u0000the Bergman space if and only if $varphi(overline{mathbb{D}^2})cap\u0000mathbb{T}^2=emptyset$ and $varphi(overline{mathbb{D}^2}setminus\u0000mathbb{T}^2)cap bmathbb{D}^2=emptyset$.","PeriodicalId":501036,"journal":{"name":"arXiv - MATH - Functional Analysis","volume":"92 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142249336","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}
The Special Affine Fourier Transformation(SAFT), which generalizes several well-known unitary transformations, has been demonstrated as a valuable tool in signal processing and optics. In this paper, we explore the multivariate dynamical sampling problem in shift-invariant spaces associated with the multi-dimensional SAFT. Specifically, we derive a sufficient and necessary condition under which a function in a shift-invariant space can be stably recovered from its dynamical sampling measurements associated with the multi-dimensional SAFT . We also present a straightforward example to elucidate our main result.
{"title":"Dynamical Sampling in Shift-Invariant Spaces Associated with multi-dimensional Special Affine Fourier Transform","authors":"Meng Ning, Li-Ping Wu, Qing-yue Zhang, Bei Liu","doi":"arxiv-2409.08506","DOIUrl":"https://doi.org/arxiv-2409.08506","url":null,"abstract":"The Special Affine Fourier Transformation(SAFT), which generalizes several\u0000well-known unitary transformations, has been demonstrated as a valuable tool in\u0000signal processing and optics. In this paper, we explore the multivariate\u0000dynamical sampling problem in shift-invariant spaces associated with the\u0000multi-dimensional SAFT. Specifically, we derive a sufficient and necessary\u0000condition under which a function in a shift-invariant space can be stably\u0000recovered from its dynamical sampling measurements associated with the\u0000multi-dimensional SAFT . We also present a straightforward example to elucidate\u0000our main result.","PeriodicalId":501036,"journal":{"name":"arXiv - MATH - Functional Analysis","volume":"20 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142249337","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 weighted weak-type $(r,r)$ estimates for operators satisfying $(r,s)$ limited-range sparse domination of $ell^q$-type. Our results contain improvements for operators satisfying limited-range and square function sparse domination. In the case of operators $T$ satisfying standard sparse form domination such as Calder'on-Zygmund operators, we provide a new and simple proof of the sharp bound $$ |T|_{L^1_w(mathbf{R}^d)rightarrow L^{1,infty}_w(mathbf{R}^d)} lesssim [w]_1(1+log [w]_{text{FW}}). $$
{"title":"Endpoint weak-type bounds beyond Calderón-Zygmund theory","authors":"Zoe Nieraeth, Cody B. Stockdale","doi":"arxiv-2409.08921","DOIUrl":"https://doi.org/arxiv-2409.08921","url":null,"abstract":"We prove weighted weak-type $(r,r)$ estimates for operators satisfying\u0000$(r,s)$ limited-range sparse domination of $ell^q$-type. Our results contain\u0000improvements for operators satisfying limited-range and square function sparse\u0000domination. In the case of operators $T$ satisfying standard sparse form\u0000domination such as Calder'on-Zygmund operators, we provide a new and simple\u0000proof of the sharp bound $$ |T|_{L^1_w(mathbf{R}^d)rightarrow L^{1,infty}_w(mathbf{R}^d)} lesssim\u0000[w]_1(1+log [w]_{text{FW}}). $$","PeriodicalId":501036,"journal":{"name":"arXiv - MATH - Functional Analysis","volume":"14 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142249344","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 article deals with functions with a prefixed range and their inclusion in Hardy and weighted Bergman spaces. This idea was originally introduced by Hansen for Hardy spaces, and it was recently taken into weighted Bergman spaces by Karafyllia and Karamanlis. In particular, we improve a theorem of Karafyllia showing that the Hardy and Bergman numbers of any given domain coincide, that is, the Hardy and weighted Bergman spaces to which a function with prefixed range belongs can be related. The main tools in the proofs are the Green function of the domain and its universal covering map.
{"title":"The Hardy number and the Bergman number of a planar domain are equal","authors":"Dimitrios Betsakos, Francisco J. Cruz-Zamorano","doi":"arxiv-2409.09150","DOIUrl":"https://doi.org/arxiv-2409.09150","url":null,"abstract":"This article deals with functions with a prefixed range and their inclusion\u0000in Hardy and weighted Bergman spaces. This idea was originally introduced by\u0000Hansen for Hardy spaces, and it was recently taken into weighted Bergman spaces\u0000by Karafyllia and Karamanlis. In particular, we improve a theorem of Karafyllia\u0000showing that the Hardy and Bergman numbers of any given domain coincide, that\u0000is, the Hardy and weighted Bergman spaces to which a function with prefixed\u0000range belongs can be related. The main tools in the proofs are the Green\u0000function of the domain and its universal covering map.","PeriodicalId":501036,"journal":{"name":"arXiv - MATH - Functional Analysis","volume":"14 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142249335","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 characterize rotund, uniformly rotund, locally uniformly rotund and compactly locally uniformly rotund spaces in terms of set of almost farthest points from the unit sphere using the generalized diameter. For this we introduce few notions involving the almost farthest points, namely strongly remotal, strongly uniquely remotal and uniformly strongly uniquely remotal sets. As a consequence, we obtain some characterizations of the aforementioned rotundity properties in terms of existing proximinality notions.
{"title":"Characterizations of some rotund properties in terms of farthest points","authors":"Arunachala Prasath C, Vamsinadh Thota","doi":"arxiv-2409.08697","DOIUrl":"https://doi.org/arxiv-2409.08697","url":null,"abstract":"We characterize rotund, uniformly rotund, locally uniformly rotund and\u0000compactly locally uniformly rotund spaces in terms of set of almost farthest\u0000points from the unit sphere using the generalized diameter. For this we\u0000introduce few notions involving the almost farthest points, namely strongly\u0000remotal, strongly uniquely remotal and uniformly strongly uniquely remotal\u0000sets. As a consequence, we obtain some characterizations of the aforementioned\u0000rotundity properties in terms of existing proximinality notions.","PeriodicalId":501036,"journal":{"name":"arXiv - MATH - Functional Analysis","volume":"33 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142268387","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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