We construct a large family of positive-definite kernels $K: mathbb{D}^ntimes mathbb{D}^n to mbox{M} (r, mathbb C)$, holomorphic in the first variable and anti-holomorphic in the second, that are quasi-invariant with respect to the subgroup $mbox{M"ob} timescdotstimes mbox{M"ob}$ ($n$ times) of the bi-holomorphic automorphism group of $mathbb{D}^n$. The adjoint of the $n$ - tuples of multiplication operators by the co-ordinate functions on the Hilbert spaces $mathcal H_K$ determined by $K$ is then homogeneous with respect to this subgroup. We show that these $n$ - tuples are irreducible, are in the Cowen-Douglas class $mathrm B_r(mathbb D^n)$ and that they are mutually pairwise unitarily inequivalent.
{"title":"A family of homogeneous operators in the Cowen–Douglas class over the poly-disc","authors":"Prahllad Deb, S. Hazra","doi":"10.4064/sm220630-10-1","DOIUrl":"https://doi.org/10.4064/sm220630-10-1","url":null,"abstract":"We construct a large family of positive-definite kernels $K: mathbb{D}^ntimes mathbb{D}^n to mbox{M} (r, mathbb C)$, holomorphic in the first variable and anti-holomorphic in the second, that are quasi-invariant with respect to the subgroup $mbox{M\"ob} timescdotstimes mbox{M\"ob}$ ($n$ times) of the bi-holomorphic automorphism group of $mathbb{D}^n$. The adjoint of the $n$ - tuples of multiplication operators by the co-ordinate functions on the Hilbert spaces $mathcal H_K$ determined by $K$ is then homogeneous with respect to this subgroup. We show that these $n$ - tuples are irreducible, are in the Cowen-Douglas class $mathrm B_r(mathbb D^n)$ and that they are mutually pairwise unitarily inequivalent.","PeriodicalId":51179,"journal":{"name":"Studia Mathematica","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-05-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41789245","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The aim of the paper is to characterize (pre)compactness in the spaces of Lipschitz/H"older continuous mappings acting from a compact metric space to a normed space. To this end some extensions and generalizations of already existing compactness criteria for the spaces of bounded and continuous mappings with values in normed spaces needed to be established. Those auxiliary results, which are interesting on their own since they use a concept of equicontinuity not seen in the literature, are based on an abstract compactness criterion related to the recently introduced notion of an equinormed set.
{"title":"Compactness in Lipschitz spaces and around","authors":"J. Gulgowski, P. Kasprzak, Piotr Ma'ckowiak","doi":"10.4064/sm221020-16-3","DOIUrl":"https://doi.org/10.4064/sm221020-16-3","url":null,"abstract":"The aim of the paper is to characterize (pre)compactness in the spaces of Lipschitz/H\"older continuous mappings acting from a compact metric space to a normed space. To this end some extensions and generalizations of already existing compactness criteria for the spaces of bounded and continuous mappings with values in normed spaces needed to be established. Those auxiliary results, which are interesting on their own since they use a concept of equicontinuity not seen in the literature, are based on an abstract compactness criterion related to the recently introduced notion of an equinormed set.","PeriodicalId":51179,"journal":{"name":"Studia Mathematica","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49121536","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
. We prove a continuous-parameter version of the recent theorem of Katznelson–Tzafiri type for power-bounded operators which have a bounded calculus for analytic Besov functions. We also show that the result can be extended to some operators which have functional calculi with respect to some larger algebras.
.我们证明了最近关于幂有界算子的Katznelson–Tza fi ri型定理的连续参数版本，该算子具有解析Besov函数的有界演算。我们还证明了这个结果可以推广到一些算子，这些算子对于一些较大的代数具有函数演算。
{"title":"A continuous-parameter Katznelson–Tzafriri theorem for algebras of analytic functions","authors":"C. Batty, D. Seifert","doi":"10.4064/sm220428-13-10","DOIUrl":"https://doi.org/10.4064/sm220428-13-10","url":null,"abstract":". We prove a continuous-parameter version of the recent theorem of Katznelson–Tzafiri type for power-bounded operators which have a bounded calculus for analytic Besov functions. We also show that the result can be extended to some operators which have functional calculi with respect to some larger algebras.","PeriodicalId":51179,"journal":{"name":"Studia Mathematica","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-04-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48185566","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Antonio Avil'es, Stefano Ciaci, Johann Langemets, A. Lissitsin, A. R. Zoca
. It is known that a Banach space contains an isomorphic copy of c 0 if, and only if, it can be equivalently renormed to be almost square. We introduce and study transfinite versions of almost square Banach spaces with the purpose to relate them to the containment of isomorphic copies of c 0 ( κ ), where κ is some uncountable cardinal. We also provide several examples and stability results of the above properties by taking direct sums, tensor products and ultraproducts. By connecting the above properties with transfinite analogues of the strong diameter two property and octahedral norms, we obtain a solution to an open question from [8].
{"title":"Transfinite almost square Banach spaces","authors":"Antonio Avil'es, Stefano Ciaci, Johann Langemets, A. Lissitsin, A. R. Zoca","doi":"10.4064/sm220517-4-11","DOIUrl":"https://doi.org/10.4064/sm220517-4-11","url":null,"abstract":". It is known that a Banach space contains an isomorphic copy of c 0 if, and only if, it can be equivalently renormed to be almost square. We introduce and study transfinite versions of almost square Banach spaces with the purpose to relate them to the containment of isomorphic copies of c 0 ( κ ), where κ is some uncountable cardinal. We also provide several examples and stability results of the above properties by taking direct sums, tensor products and ultraproducts. By connecting the above properties with transfinite analogues of the strong diameter two property and octahedral norms, we obtain a solution to an open question from [8].","PeriodicalId":51179,"journal":{"name":"Studia Mathematica","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-04-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47570666","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
For $X(n)$ a Steinhaus random multiplicative function, we study the maximal size of the random Dirichlet polynomial $$ D_N(t) = frac1{sqrt{N}} sum_{n leq N} X(n) n^{it}, $$ with $t$ in various ranges. In particular, for fixed $C>0$ and any small $varepsilon>0$ we show that, with high probability, $$ exp( (log N)^{1/2-varepsilon} ) ll sup_{|t| leq N^C} |D_N(t)| ll exp( (log N)^{1/2+varepsilon}). $$
{"title":"Extremal bounds for Dirichlet polynomials with random multiplicative coefficients","authors":"Jacques Benatar, Alon Nishry","doi":"10.4064/sm220829-6-3","DOIUrl":"https://doi.org/10.4064/sm220829-6-3","url":null,"abstract":"For $X(n)$ a Steinhaus random multiplicative function, we study the maximal size of the random Dirichlet polynomial $$ D_N(t) = frac1{sqrt{N}} sum_{n leq N} X(n) n^{it}, $$ with $t$ in various ranges. In particular, for fixed $C>0$ and any small $varepsilon>0$ we show that, with high probability, $$ exp( (log N)^{1/2-varepsilon} ) ll sup_{|t| leq N^C} |D_N(t)| ll exp( (log N)^{1/2+varepsilon}). $$","PeriodicalId":51179,"journal":{"name":"Studia Mathematica","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-04-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41816736","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
With the aid of a Gromov hyperbolic characterization of uniform domains, we first give an affirmative answer to an open question arisen by Väisälä under weaker assumption. Next, we show that the three-point condition introduced by Väisälä is necessary to obtain quasisymmetry for quasimöbius maps between bounded connected spaces in a quantitative way. Based on these two results, we investigate the boundary behavior of freely quasiconformal and quasihyperbolic mappings on uniform domains of Banach spaces and partially answer another question raised by Väisälä in different ways.
{"title":"Gromov hyperbolicity in the free quasiworld. I","authors":"Qingshan Zhou, S. Ponnusamy","doi":"10.4064/sm210825-7-3","DOIUrl":"https://doi.org/10.4064/sm210825-7-3","url":null,"abstract":"With the aid of a Gromov hyperbolic characterization of uniform domains, we first give an affirmative answer to an open question arisen by Väisälä under weaker assumption. Next, we show that the three-point condition introduced by Väisälä is necessary to obtain quasisymmetry for quasimöbius maps between bounded connected spaces in a quantitative way. Based on these two results, we investigate the boundary behavior of freely quasiconformal and quasihyperbolic mappings on uniform domains of Banach spaces and partially answer another question raised by Väisälä in different ways.","PeriodicalId":51179,"journal":{"name":"Studia Mathematica","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43477765","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study long chains of iterated weak∗ derived sets, that is sets of all weak∗ limits of bounded nets, of subspaces with the additional property that the penultimate weak∗ derived set is a proper norm dense subspace of the dual. We extend the result of Ostrovskii and show, that in the dual of any nonquasi-reflexive Banach space containing an infinite-dimensional subspace with separable dual, we can find for any countable successor ordinal α a subspace, whose weak∗ derived set of order α is proper and norm dense.
{"title":"On subspaces whose weak$^*$ derived sets are proper and norm dense","authors":"Zdenvek Silber","doi":"10.4064/sm220303-29-4","DOIUrl":"https://doi.org/10.4064/sm220303-29-4","url":null,"abstract":"We study long chains of iterated weak∗ derived sets, that is sets of all weak∗ limits of bounded nets, of subspaces with the additional property that the penultimate weak∗ derived set is a proper norm dense subspace of the dual. We extend the result of Ostrovskii and show, that in the dual of any nonquasi-reflexive Banach space containing an infinite-dimensional subspace with separable dual, we can find for any countable successor ordinal α a subspace, whose weak∗ derived set of order α is proper and norm dense.","PeriodicalId":51179,"journal":{"name":"Studia Mathematica","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47106082","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let $mathbf{X}={X_t}_{tgeq 0}$ be a L'evy process in $mathbb{R}^d$ and $Omega$ be an open subset of $mathbb{R}^d$ with finite Lebesgue measure. In this article we consider the quantity $H(t)=int_{Omega} mathbb{P}^x (X_tinOmega^c) , mathrm{d}x$ which is called the heat content. We study its asymptotic expansion for isotropic $alpha$-stable L'evy processes and more general L'evy processes, under mild assumptions on the characteristic exponent.
{"title":"Asymptotic expansion of the nonlocal heat content","authors":"T. Grzywny, Julia Lenczewska","doi":"10.4064/sm220831-26-1","DOIUrl":"https://doi.org/10.4064/sm220831-26-1","url":null,"abstract":"Let $mathbf{X}={X_t}_{tgeq 0}$ be a L'evy process in $mathbb{R}^d$ and $Omega$ be an open subset of $mathbb{R}^d$ with finite Lebesgue measure. In this article we consider the quantity $H(t)=int_{Omega} mathbb{P}^x (X_tinOmega^c) , mathrm{d}x$ which is called the heat content. We study its asymptotic expansion for isotropic $alpha$-stable L'evy processes and more general L'evy processes, under mild assumptions on the characteristic exponent.","PeriodicalId":51179,"journal":{"name":"Studia Mathematica","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-02-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44891899","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove rigidity and classification results for type III factors given by nonsingular Bernoulli actions of the free groups and more general free product groups. This includes a large family of nonisomorphic Bernoulli crossed products of type III$_1$ that cannot be distinguished by Connes $tau$-invariant. These are the first such classification results beyond the well studied probability measure preserving case.
{"title":"Classification results for\u0000nonsingular Bernoulli crossed products","authors":"S. Vaes, Bram Verjans","doi":"10.4064/sm220217-11-5","DOIUrl":"https://doi.org/10.4064/sm220217-11-5","url":null,"abstract":"We prove rigidity and classification results for type III factors given by nonsingular Bernoulli actions of the free groups and more general free product groups. This includes a large family of nonisomorphic Bernoulli crossed products of type III$_1$ that cannot be distinguished by Connes $tau$-invariant. These are the first such classification results beyond the well studied probability measure preserving case.","PeriodicalId":51179,"journal":{"name":"Studia Mathematica","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43619717","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
ABSTRACT. Consider a mixing dynamical systems ([0, 1], T, μ), for instance a piecewise expanding interval map with a Gibbs measure μ. Given a non-summable sequence (mk) of non-negative numbers, one may define rk(x) such that μ(B(x, rk(x)) = mk. It is proved that for almost all x, the number of k ≤ n such that Tk(x) ∈ Bk(x) is approximately equal to m1 + . . . + mn. This is a sort of strong Borel–Cantelli lemma for recurrence. A consequence is that
{"title":"A strong Borel–Cantelli lemma for recurrence","authors":"T. Persson","doi":"10.4064/sm220216-2-7","DOIUrl":"https://doi.org/10.4064/sm220216-2-7","url":null,"abstract":"ABSTRACT. Consider a mixing dynamical systems ([0, 1], T, μ), for instance a piecewise expanding interval map with a Gibbs measure μ. Given a non-summable sequence (mk) of non-negative numbers, one may define rk(x) such that μ(B(x, rk(x)) = mk. It is proved that for almost all x, the number of k ≤ n such that Tk(x) ∈ Bk(x) is approximately equal to m1 + . . . + mn. This is a sort of strong Borel–Cantelli lemma for recurrence. A consequence is that","PeriodicalId":51179,"journal":{"name":"Studia Mathematica","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42231817","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","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：