Let $(X,(p_j))$ be a Frechet space with a Schauder basis and without continuous norm, where $(p_j)$ is an increasing sequence of seminorms inducing the topology of $X$. We show that $X$ satisfies the Invariant Subspace Property if and only if there exists $j_0ge 1$ such that $ker p_{j+1}$ is of finite codimension in $ker p_{j}$ for every $jge j_0$.
{"title":"Invariant subspaces for Fréchet spaces without continuous norm","authors":"Q. Menet","doi":"10.1090/proc/15418","DOIUrl":"https://doi.org/10.1090/proc/15418","url":null,"abstract":"Let $(X,(p_j))$ be a Frechet space with a Schauder basis and without continuous norm, where $(p_j)$ is an increasing sequence of seminorms inducing the topology of $X$. We show that $X$ satisfies the Invariant Subspace Property if and only if there exists $j_0ge 1$ such that $ker p_{j+1}$ is of finite codimension in $ker p_{j}$ for every $jge j_0$.","PeriodicalId":8426,"journal":{"name":"arXiv: Functional Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-07-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86992587","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 $X$ be a Banach space. Then $X$ is complemented in the bidual $X^{**}$ if and only if there exists an invariant mean $ell_infty(G, X)to X$ with respect to a free Abelian group $G$ of rank equal to the cardinality of $X^{**}$, and this happens if and only if there exists an invariant mean with respect to the additive group of $X^{**}$. This improves upon previous results due to Bustos Domecq =and the second-named author, where certain idempotent semigroups of cardinality equal to the cardinality of $X^{**}$ were considered, and answers a question of J.M.F. Castillo (private communication). En route to the proof of the main result, we endow the family of all finite-dimensional subspaces of an infinite-dimensional vector space with a structure of a free commutative monoid with the property that the product of two subspaces contains the respective subspaces, which is possibly of interest in itself.
{"title":"Invariant means on Abelian groups capture complementability of Banach spaces in their second duals","authors":"Adam P. Goucher, Tomasz Kania","doi":"10.4064/SM200706-15-1","DOIUrl":"https://doi.org/10.4064/SM200706-15-1","url":null,"abstract":"Let $X$ be a Banach space. Then $X$ is complemented in the bidual $X^{**}$ if and only if there exists an invariant mean $ell_infty(G, X)to X$ with respect to a free Abelian group $G$ of rank equal to the cardinality of $X^{**}$, and this happens if and only if there exists an invariant mean with respect to the additive group of $X^{**}$. This improves upon previous results due to Bustos Domecq =and the second-named author, where certain idempotent semigroups of cardinality equal to the cardinality of $X^{**}$ were considered, and answers a question of J.M.F. Castillo (private communication). En route to the proof of the main result, we endow the family of all finite-dimensional subspaces of an infinite-dimensional vector space with a structure of a free commutative monoid with the property that the product of two subspaces contains the respective subspaces, which is possibly of interest in itself.","PeriodicalId":8426,"journal":{"name":"arXiv: Functional Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-07-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81788095","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}
A decomposition theorem for self-adjoint operators proved by Riesz and Lorch is extended to normal operators. This extension gives a new proof of the spectral theorem for unbounded normal operators.
{"title":"DECOMPOSITION OF NORMAL OPERATORS AND ITS APPLICATION TO SPECTRAL THEOREM","authors":"Katsukuni Nakagawa","doi":"10.17654/FA012010037","DOIUrl":"https://doi.org/10.17654/FA012010037","url":null,"abstract":"A decomposition theorem for self-adjoint operators proved by Riesz and Lorch is extended to normal operators. This extension gives a new proof of the spectral theorem for unbounded normal operators.","PeriodicalId":8426,"journal":{"name":"arXiv: Functional Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-06-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77075006","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 concept of asymptotically nonexpansive mappings is an important generalization of the class of nonexpansive mappings. Implicit midpoint procedures are extremely fundamental for solving equations involving nonlinear operators. This paper studies the convergence analysis of the class of asymptotically nonexpansive mappings by the implicit midpoint iterative procedures. The necessary conditions for the convergence of the class of asymptotically nonexpansive mappings are established, by using a well-known iterative algorithm which plays important roles in the computation of fixed points of nonlinear mappings. A numerical example is presented to illustrate the convergence result. Under relaxed conditions on the parameters, some algorithms and strong convergence results were derived to obtain some results in the literature as corollaries.
{"title":"The Implicit Midpoint Procedures for Asymptotically Nonexpansive Mappings","authors":"M. Aibinu, S. C. Thakur, S. Moyo","doi":"10.1155/2020/6876385","DOIUrl":"https://doi.org/10.1155/2020/6876385","url":null,"abstract":"The concept of asymptotically nonexpansive mappings is an important generalization of the class of nonexpansive mappings. Implicit midpoint procedures are extremely fundamental for solving equations involving nonlinear operators. This paper studies the convergence analysis of the class of asymptotically nonexpansive mappings by the implicit midpoint iterative procedures. The necessary conditions for the convergence of the class of asymptotically nonexpansive mappings are established, by using a well-known iterative algorithm which plays important roles in the computation of fixed points of nonlinear mappings. A numerical example is presented to illustrate the convergence result. Under relaxed conditions on the parameters, some algorithms and strong convergence results were derived to obtain some results in the literature as corollaries.","PeriodicalId":8426,"journal":{"name":"arXiv: Functional Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83370531","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}
Pub Date : 2020-06-05DOI: 10.33581/2520-6508-2020-2-28-35
A. Mirotin
Hausdorff operators on the real line and multidimensional Euclidean spaces originated from some classical summation methods. Now it is an active research area. Hausdorff operators on general groups were defined and studied by the author since 2019. The purpose of this paper is to define and study Hausdorff operators on Lebesgue and real Hardy spaces over homogeneous spaces of locally compact groups. We introduce in particular an atomic Hardy space over homogeneous spaces of locally compact groups and obtain conditions for boundedness of Hausdorff operators on such spaces. Several corollaries are considered and unsolved problems are formulated.
{"title":"Hausdorff operators on homogeneous spaces of locally compact groups","authors":"A. Mirotin","doi":"10.33581/2520-6508-2020-2-28-35","DOIUrl":"https://doi.org/10.33581/2520-6508-2020-2-28-35","url":null,"abstract":"Hausdorff operators on the real line and multidimensional Euclidean spaces originated from some classical summation methods. Now it is an active research area. Hausdorff operators on general groups were defined and studied by the author since 2019. The purpose of this paper is to define and study Hausdorff operators on Lebesgue and real Hardy spaces over homogeneous spaces of locally compact groups. We introduce in particular an atomic Hardy space over homogeneous spaces of locally compact groups and obtain conditions for boundedness of Hausdorff operators on such spaces. Several corollaries are considered and unsolved problems are formulated.","PeriodicalId":8426,"journal":{"name":"arXiv: Functional Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82215482","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 extend to Banach space nest algebras the theory of essential supports and support function pairs of their bimodules, thereby obtaining Banach space counterparts of long established results for Hilbert space nest algebras. Namely, given a Banach space nest algebra $mathcal A$, we charaterise the maximal and the minimal $mathcal A$-bimodules having a given essential support function or support function pair. These characterisations are complete except for the minimal $mathcal A$-bimodule corresponding to a support function pair, in which case we make some headway. We also show that the weakly closed bimodules of a Banach space nest algebra are exactly those that are reflexive operator spaces. To this end, we crucially prove that reflexive bimodules determine uniquely a certain class of admissible support functions.
将其双模的基本支持理论及其支持函数对推广到Banach空间巢代数中，从而得到Hilbert空间巢代数长期建立结果的Banach空间对应物。也就是说，给定一个Banach空间巢代数$mathcal a $，我们刻画了具有给定本质支持函数或支持函数对的最大和最小$mathcal a $-双模。除了与支持函数对对应的最小的数学A -双模之外，这些特征都是完整的，在这种情况下，我们取得了一些进展。我们还证明了Banach空间巢代数的弱闭双模正是那些自反算子空间。为此，我们关键地证明了自反双模唯一地决定了一类可容许的支持函数。
{"title":"Bimodules of Banach Space Nest Algebras","authors":"Lu'is Duarte, L. Oliveira","doi":"10.1093/QMATH/HAAB028","DOIUrl":"https://doi.org/10.1093/QMATH/HAAB028","url":null,"abstract":"We extend to Banach space nest algebras the theory of essential supports and support function pairs of their bimodules, thereby obtaining Banach space counterparts of long established results for Hilbert space nest algebras. Namely, given a Banach space nest algebra $mathcal A$, we charaterise the maximal and the minimal $mathcal A$-bimodules having a given essential support function or support function pair. These characterisations are complete except for the minimal $mathcal A$-bimodule corresponding to a support function pair, in which case we make some headway. \u0000We also show that the weakly closed bimodules of a Banach space nest algebra are exactly those that are reflexive operator spaces. To this end, we crucially prove that reflexive bimodules determine uniquely a certain class of admissible support functions.","PeriodicalId":8426,"journal":{"name":"arXiv: Functional Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90974912","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 some metric spaces of self-adjoint operators, it is shown that the set of operators whose spectral measures have simultaneously zero upper-Hausdorff and one lower-packing dimensions contains a dense $G_delta$ subset. Applications include sets of limit-periodic operators.
{"title":"Generic zero-Hausdorff and one-packing spectral measures","authors":"S. L. Carvalho, C. R. de Oliveira","doi":"10.1063/1.5141763","DOIUrl":"https://doi.org/10.1063/1.5141763","url":null,"abstract":"For some metric spaces of self-adjoint operators, it is shown that the set of operators whose spectral measures have simultaneously zero upper-Hausdorff and one lower-packing dimensions contains a dense $G_delta$ subset. Applications include sets of limit-periodic operators.","PeriodicalId":8426,"journal":{"name":"arXiv: Functional Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73446069","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}
Pub Date : 2020-06-01DOI: 10.1016/J.JFA.2021.109197
N. Holighaus, F. Voigtlaender
{"title":"Schur-type Banach modules of integral kernels acting on mixed-norm Lebesgue spaces","authors":"N. Holighaus, F. Voigtlaender","doi":"10.1016/J.JFA.2021.109197","DOIUrl":"https://doi.org/10.1016/J.JFA.2021.109197","url":null,"abstract":"","PeriodicalId":8426,"journal":{"name":"arXiv: Functional Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83214236","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 build up a criteria for fractional Orlicz-Sobolev extension and imbedding domains on Ahlfors $n$-regular domains.
本文建立了分数阶Orlicz-Sobolev扩展和嵌入域在Ahlfors $n$正则域上的判据。
{"title":"Fractional Orlicz–Sobolev Extension/Imbedding on Ahlfors $n$-Regular Domains","authors":"Tian Liang","doi":"10.4171/zaa/1659","DOIUrl":"https://doi.org/10.4171/zaa/1659","url":null,"abstract":"In this paper we build up a criteria for fractional Orlicz-Sobolev extension and imbedding domains on Ahlfors $n$-regular domains.","PeriodicalId":8426,"journal":{"name":"arXiv: Functional Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87375825","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 $B_{x}subseteqmathbb{R}^{n}$ denote the Euclidean ball with diameter $[0,x]$, i.e. with with center at $frac{x}{2}$ and radius $frac{left|xright|}{2}$. We call such a ball a petal. A flower $F$ is any union of petals, i.e. $F=bigcup_{xin A}B_{x}$ for any set $Asubseteqmathbb{R}^{n}$. We showed in previous work that the family of all flowers $mathcal{F}$ is in 1-1 correspondence with $mathcal{K}_{0}$ - the family of all convex bodies containing $0$. Actually, there are two essentially different such correspondences. We demonstrate a number of different non-linear constructions on $mathcal{F}$ and $mathcal{K}_{0}$. Towards this goal we further develop the theory of flowers.
{"title":"Novel View on Classical Convexity Theory","authors":"V. Milman, Liran Rotem","doi":"10.15407/mag16.03.291","DOIUrl":"https://doi.org/10.15407/mag16.03.291","url":null,"abstract":"Let $B_{x}subseteqmathbb{R}^{n}$ denote the Euclidean ball with diameter $[0,x]$, i.e. with with center at $frac{x}{2}$ and radius $frac{left|xright|}{2}$. We call such a ball a petal. A flower $F$ is any union of petals, i.e. $F=bigcup_{xin A}B_{x}$ for any set $Asubseteqmathbb{R}^{n}$. We showed in previous work that the family of all flowers $mathcal{F}$ is in 1-1 correspondence with $mathcal{K}_{0}$ - the family of all convex bodies containing $0$. Actually, there are two essentially different such correspondences. We demonstrate a number of different non-linear constructions on $mathcal{F}$ and $mathcal{K}_{0}$. Towards this goal we further develop the theory of flowers.","PeriodicalId":8426,"journal":{"name":"arXiv: Functional Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-05-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90169784","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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