In this paper we give complete descriptions of the set of square roots of certain classical operators, often providing specific formulas. The classical operators included in this discussion are the square of the unilateral shift, the Volterra operator, certain compressed shifts, the unilateral shift plus its adjoint, the Hilbert matrix, and the Cesàro operator.
{"title":"The square roots of some classical operators","authors":"J. Mashreghi, M. Ptak, W. Ross","doi":"10.4064/sm210928-19-9","DOIUrl":"https://doi.org/10.4064/sm210928-19-9","url":null,"abstract":"In this paper we give complete descriptions of the set of square roots of certain classical operators, often providing specific formulas. The classical operators included in this discussion are the square of the unilateral shift, the Volterra operator, certain compressed shifts, the unilateral shift plus its adjoint, the Hilbert matrix, and the Cesàro operator.","PeriodicalId":51179,"journal":{"name":"Studia Mathematica","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-09-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48656261","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 provide necessary and sufficient conditions for the existence of idempotents of arbitrarily large norms in the Fourier algebra A(G) and the Fourier-Stieltjes algebra B(G) of a locally compact group G. We prove that the existence of idempotents of arbitrarily large norm in B(G) implies the existence of homomorphisms of arbitrarily large norm from A(H) into B(G) for every locally compact group H. A partial converse is also obtained: the existence of homomorphisms of arbitrarily large norm from A(H) into B(G) for some amenable locally compact group H implies the existence of idempotents of arbitrarily large norm in B(G).
{"title":"Idempotents of large norm and homomorphisms of Fourier algebras","authors":"M. Anoussis, G. Eleftherakis, A. Katavolos","doi":"10.4064/sm220111-20-1","DOIUrl":"https://doi.org/10.4064/sm220111-20-1","url":null,"abstract":"We provide necessary and sufficient conditions for the existence of idempotents of arbitrarily large norms in the Fourier algebra A(G) and the Fourier-Stieltjes algebra B(G) of a locally compact group G. We prove that the existence of idempotents of arbitrarily large norm in B(G) implies the existence of homomorphisms of arbitrarily large norm from A(H) into B(G) for every locally compact group H. A partial converse is also obtained: the existence of homomorphisms of arbitrarily large norm from A(H) into B(G) for some amenable locally compact group H implies the existence of idempotents of arbitrarily large norm in B(G).","PeriodicalId":51179,"journal":{"name":"Studia Mathematica","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48035097","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}
Laura Manvcinska, V. Paulsen, I. Todorov, A. Winter
We show that the *-algebra of the product of two synchronous games is the tensor product of the corresponding *-algebras. We prove that the product game has a perfect C*-strategy if and only if each of the individual games does, and that in this case the C*-algebra of the product game is *-isomorphic to the maximal C*-tensor product of the individual C*-algebras. We provide examples of synchronous games whose synchronous values are strictly supermultiplicative.
{"title":"Products of synchronous games","authors":"Laura Manvcinska, V. Paulsen, I. Todorov, A. Winter","doi":"10.4064/sm221201-19-4","DOIUrl":"https://doi.org/10.4064/sm221201-19-4","url":null,"abstract":"We show that the *-algebra of the product of two synchronous games is the tensor product of the corresponding *-algebras. We prove that the product game has a perfect C*-strategy if and only if each of the individual games does, and that in this case the C*-algebra of the product game is *-isomorphic to the maximal C*-tensor product of the individual C*-algebras. We provide examples of synchronous games whose synchronous values are strictly supermultiplicative.","PeriodicalId":51179,"journal":{"name":"Studia Mathematica","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46337574","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 notion of Bohr chaos was introduced in [3, 4]. We answer a question raised in [3] of whether a non uniquely ergodic minimal system of positive topological entropy can be Bohr chaotic. We also prove that all systems with the specification property are Bohr chaotic, and by this line of thought give an independent proof (and stengthening) of theorem 1 of [3] for the case of invertible systems. In addition, we present an obstruction for Bohr chaos: a system with fewer than a continuum of ergodic invariant probability measures cannot be Bohr chaotic.
{"title":"Some remarks on the notion of Bohr chaos and invariant measures","authors":"Matan Tal","doi":"10.4064/sm230103-13-5","DOIUrl":"https://doi.org/10.4064/sm230103-13-5","url":null,"abstract":"The notion of Bohr chaos was introduced in [3, 4]. We answer a question raised in [3] of whether a non uniquely ergodic minimal system of positive topological entropy can be Bohr chaotic. We also prove that all systems with the specification property are Bohr chaotic, and by this line of thought give an independent proof (and stengthening) of theorem 1 of [3] for the case of invertible systems. In addition, we present an obstruction for Bohr chaos: a system with fewer than a continuum of ergodic invariant probability measures cannot be Bohr chaotic.","PeriodicalId":51179,"journal":{"name":"Studia Mathematica","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49254941","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 { s n } n ∈ N be a decreasing nonsummable sequence of positive reals. In this paper, we investigate the weighted Birkhoff average 1 S n P n − 1 k =0 s k φ ( T k x ) on aperiodic irreducible subshift of finite type Σ A where φ : Σ A 7→ R is a continuous potential. Firstly, we show the entropy spectrum of the weighed Birkhoff averages remains the same as that of the Birkhoff averages. Then we cal-culate the packing spectrum of the weighed Birkhoff averages. It turns out that we can have two cases, either the packing dimension of every level set equals to its Hausdorff dimension or for every nonempty level set it is equal to the packing dimension of the whole space.
。设{s n} n∈n是一个递减的不可和的正实数序列。本文研究了有限型Σ A的非周期不可约子位移上的加权Birkhoff平均1 S n P n−1 k =0 S k φ (T k x)，其中φ: Σ a7→R为连续势。首先，我们证明加权伯克霍夫平均的熵谱与伯克霍夫平均的熵谱保持一致。然后计算加权Birkhoff平均的堆积谱。我们可以有两种情况，要么每个水平集的填充维数等于它的豪斯多夫维数，要么每个非空水平集的填充维数等于整个空间的填充维数。
{"title":"Spectrum of weighted Birkhoff average","authors":"Bal'azs B'ar'any, M. Rams, Ruxi Shi","doi":"10.4064/sm210908-19-6","DOIUrl":"https://doi.org/10.4064/sm210908-19-6","url":null,"abstract":". Let { s n } n ∈ N be a decreasing nonsummable sequence of positive reals. In this paper, we investigate the weighted Birkhoff average 1 S n P n − 1 k =0 s k φ ( T k x ) on aperiodic irreducible subshift of finite type Σ A where φ : Σ A 7→ R is a continuous potential. Firstly, we show the entropy spectrum of the weighed Birkhoff averages remains the same as that of the Birkhoff averages. Then we cal-culate the packing spectrum of the weighed Birkhoff averages. It turns out that we can have two cases, either the packing dimension of every level set equals to its Hausdorff dimension or for every nonempty level set it is equal to the packing dimension of the whole space.","PeriodicalId":51179,"journal":{"name":"Studia Mathematica","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46131756","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 deal with a global Calder'on-Zygmund type estimate for elliptic obstacle problems of $p$-Laplacian type with measure data. For this paper, we focus on the singular case of growth exponent, i.e. $1
{"title":"Calderón–Zygmund type estimates for singular quasilinear elliptic obstacle problems with measure data","authors":"Minh-Phuong Tran, Thanh-Nhan Nguyen, Phuoc-Nguyen Huynh","doi":"10.4064/sm220321-26-4","DOIUrl":"https://doi.org/10.4064/sm220321-26-4","url":null,"abstract":"We deal with a global Calder'on-Zygmund type estimate for elliptic obstacle problems of $p$-Laplacian type with measure data. For this paper, we focus on the singular case of growth exponent, i.e. $1<p le 2-frac{1}{n}$. In addition, the emphasis of this paper is in obtaining the Lorentz bounds for the gradient of solutions with the use of fractional maximal operators.","PeriodicalId":51179,"journal":{"name":"Studia Mathematica","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46719388","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 consider a possibly degenerate Kolmogorov-Ornstein-Uhlenbeck operator of the form L = Tr(BD2) + 〈Az, D〉, where A, B are N × N matrices, z ∈ RN , N ≥ 1, which satisfy the Kalman condition which is equivalent to the hypoellipticity condition. We prove the following stability result: the Schauder and Sobolev estimates associated with the corresponding parabolic Cauchy problem remain valid, with the same constant, for the parabolic Cauchy problem associated with a second order perturbation of L, namely for L + Tr(S(t)D2) where S(t) is a non-negative definite N × N matrix depending continuously on t ∈ [0, T ]. Our approach relies on the perturbative technique based on the Poisson process introduced in [15].
{"title":"Poisson process and sharp constants in $L^p$ and Schauder estimates for a class of degenerate Kolmogorov operators","authors":"L. Marino, S. Menozzi, E. Priola","doi":"10.4064/sm210819-13-4","DOIUrl":"https://doi.org/10.4064/sm210819-13-4","url":null,"abstract":"We consider a possibly degenerate Kolmogorov-Ornstein-Uhlenbeck operator of the form L = Tr(BD2) + 〈Az, D〉, where A, B are N × N matrices, z ∈ RN , N ≥ 1, which satisfy the Kalman condition which is equivalent to the hypoellipticity condition. We prove the following stability result: the Schauder and Sobolev estimates associated with the corresponding parabolic Cauchy problem remain valid, with the same constant, for the parabolic Cauchy problem associated with a second order perturbation of L, namely for L + Tr(S(t)D2) where S(t) is a non-negative definite N × N matrix depending continuously on t ∈ [0, T ]. Our approach relies on the perturbative technique based on the Poisson process introduced in [15].","PeriodicalId":51179,"journal":{"name":"Studia Mathematica","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-07-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45119558","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 give a sufficient condition for a Banach space with which the homogeneous extension of a surjective isometry from the unit sphere of it onto another one is real-linear. The condition is satisfied by a uniform algebra and a certain extremely $C$-regular space of real-valued continuous functions.
{"title":"The Mazur–Ulam property for uniform algebras","authors":"O. Hatori","doi":"10.4064/sm210703-11-9","DOIUrl":"https://doi.org/10.4064/sm210703-11-9","url":null,"abstract":"We give a sufficient condition for a Banach space with which the homogeneous extension of a surjective isometry from the unit sphere of it onto another one is real-linear. The condition is satisfied by a uniform algebra and a certain extremely $C$-regular space of real-valued continuous functions.","PeriodicalId":51179,"journal":{"name":"Studia Mathematica","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43880612","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 (Xj , dj , μj), j = 0, 1, . . . ,m be metric measure spaces. Given 0 < pκ ≤ ∞ for κ = 1, . . . ,m and an analytic family of multilinear operators Tz : L 1(X1)× · · ·L m (Xm)→ Lloc(X0), for z in the complex unit strip, we prove a theorem in the spirit of Stein’s complex interpolation for analytic families. Analyticity and our admissibility condition are defined in the weak (integral) sense and relax the pointwise definitions given in [9]. Continuous functions with compact support are natural dense subspaces of Lebesgue spaces over metric measure spaces and we assume the operators Tz are initially defined on them. Our main lemma concerns the approximation of continuous functions with compact support by similar functions that depend analytically in an auxiliary parameter z. An application of the main theorem concerning bilinear estimates for Schrödinger operators on Lp is included.
设(Xj, dj， μj)， j = 0,1，…，m是度量空间。给定0 < pκ≤∞，对于κ = 1，…，m和多元线性算子族解析Tz: L 1(X1)×···L m (Xm)→Lloc(X0)，对于复单元带中的z，我们以Stein复插值的精神证明了一个定理。在弱(积分)意义上定义了解析性和容许条件，放宽了[9]中给出的逐点定义。具有紧支持的连续函数是勒贝格空间在度量度量空间上的自然稠密子空间，我们假设算子Tz在其上是初始定义的。我们的主要引理涉及由解析依赖于辅助参数z的相似函数逼近具有紧支持的连续函数。包括Lp上Schrödinger算子双线性估计的主要定理的一个应用。
{"title":"Interpolation for analytic families of multilinear operators on metric measure spaces","authors":"L. Grafakos, E. Ouhabaz","doi":"10.4064/sm210630-11-1","DOIUrl":"https://doi.org/10.4064/sm210630-11-1","url":null,"abstract":"Let (Xj , dj , μj), j = 0, 1, . . . ,m be metric measure spaces. Given 0 < pκ ≤ ∞ for κ = 1, . . . ,m and an analytic family of multilinear operators Tz : L 1(X1)× · · ·L m (Xm)→ Lloc(X0), for z in the complex unit strip, we prove a theorem in the spirit of Stein’s complex interpolation for analytic families. Analyticity and our admissibility condition are defined in the weak (integral) sense and relax the pointwise definitions given in [9]. Continuous functions with compact support are natural dense subspaces of Lebesgue spaces over metric measure spaces and we assume the operators Tz are initially defined on them. Our main lemma concerns the approximation of continuous functions with compact support by similar functions that depend analytically in an auxiliary parameter z. An application of the main theorem concerning bilinear estimates for Schrödinger operators on Lp is included.","PeriodicalId":51179,"journal":{"name":"Studia Mathematica","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48909396","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 consider the weighted Sobolev spaces associated with nonisotropic dilations of Calderón-Torchinsky and characterize the spaces by the square functions of Marcinkiewicz type including those defined with repeated uses of averaging operation.
{"title":"Sobolev spaces with non-isotropic dilations and\u0000square functions of Marcinkiewicz type","authors":"Shuichi Sato","doi":"10.4064/sm210819-19-3","DOIUrl":"https://doi.org/10.4064/sm210819-19-3","url":null,"abstract":"We consider the weighted Sobolev spaces associated with nonisotropic dilations of Calderón-Torchinsky and characterize the spaces by the square functions of Marcinkiewicz type including those defined with repeated uses of averaging operation.","PeriodicalId":51179,"journal":{"name":"Studia Mathematica","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2021-06-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43791876","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}
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