In the signal-processing literature, a frame is a mechanism for performing analysis and reconstruction in a Hilbert space. By contrast, in quantum theory, a positive operator-valued measure (POVM) decomposes a Hilbert-space vector for the purpose of computing measurement probabilities. Frames and their most common generalizations can be seen to give rise to POVMs, but does every reasonable POVM arise from a type of frame? In this paper we answer this question using a Radon-Nikodym-type result.
{"title":"Positive operator-valued measures and densely defined operator-valued frames","authors":"B. Robinson, Bill Moran, D. Cochran","doi":"10.1216/RMJ.2021.51.265","DOIUrl":"https://doi.org/10.1216/RMJ.2021.51.265","url":null,"abstract":"In the signal-processing literature, a frame is a mechanism for performing analysis and reconstruction in a Hilbert space. By contrast, in quantum theory, a positive operator-valued measure (POVM) decomposes a Hilbert-space vector for the purpose of computing measurement probabilities. Frames and their most common generalizations can be seen to give rise to POVMs, but does every reasonable POVM arise from a type of frame? In this paper we answer this question using a Radon-Nikodym-type result.","PeriodicalId":8426,"journal":{"name":"arXiv: Functional Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80509956","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 introduce the statistically multiplicative convergent sequences in locally solid Riesz algebras with respect to the algebra multiplication and the solid topology. We study on this concept and we give the notion of $mathbb{st_m}$-bounded sequence, and also, we prove some relations between this convergence and the other convergences such as the order convergence and the statistical convergence in topological spaces. Also, we give some results related to semiprime $f$-algebras.
{"title":"Statistically multiplicative convergence on locally solid Riesz algebras","authors":"A. Aydın, M. Et","doi":"10.3906/mat-2102-20","DOIUrl":"https://doi.org/10.3906/mat-2102-20","url":null,"abstract":"In this paper, we introduce the statistically multiplicative convergent sequences in locally solid Riesz algebras with respect to the algebra multiplication and the solid topology. We study on this concept and we give the notion of $mathbb{st_m}$-bounded sequence, and also, we prove some relations between this convergence and the other convergences such as the order convergence and the statistical convergence in topological spaces. Also, we give some results related to semiprime $f$-algebras.","PeriodicalId":8426,"journal":{"name":"arXiv: Functional Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-04-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83872801","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 show that the typical nonexpansive mapping on a small enough subset of a CAT($kappa$)-space is a contraction in the sense of Rakotch. By typical we mean that the set of nonexpansive mapppings without this property is a $sigma$-porous set and therefore also of the first Baire category. Moreover, we exhibit metric spaces where strict contractions are not dense in the space of nonexpansive mappings. In some of these cases we show that all continuous self-mappings have a fixed point nevertheless.
{"title":"On the existence of fixed points for typical nonexpansive mappings on spaces with positive curvature","authors":"C. Bargetz, Michael Dymond, Emir Medjic, S. Reich","doi":"10.12775/TMNA.2020.040","DOIUrl":"https://doi.org/10.12775/TMNA.2020.040","url":null,"abstract":"We show that the typical nonexpansive mapping on a small enough subset of a CAT($kappa$)-space is a contraction in the sense of Rakotch. By typical we mean that the set of nonexpansive mapppings without this property is a $sigma$-porous set and therefore also of the first Baire category. Moreover, we exhibit metric spaces where strict contractions are not dense in the space of nonexpansive mappings. In some of these cases we show that all continuous self-mappings have a fixed point nevertheless.","PeriodicalId":8426,"journal":{"name":"arXiv: Functional Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-04-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75438608","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, $Ain B(X)$ and $M$ be an invariant subspace of $A$. We present an alternative proof that, if the spectrum of the restriction of $A$ to $M$ contains a point that is in any given hole in the spectrum of $A$, then the entire hole is in the spectrum of the restriction.
设$X$是一个巴拿赫空间，$ a in B(X)$， $M$是$ a $的不变子空间。我们给出了另一种证明，如果$A$到$M$的限制的谱中包含一个点，该点在$A$的谱中的任何给定洞中，则整个洞都在该限制的谱中。
{"title":"The spectrum of the restriction to an invariant subspace","authors":"D. Drivaliaris, N. Yannakakis","doi":"10.7153/oam-2020-14-19","DOIUrl":"https://doi.org/10.7153/oam-2020-14-19","url":null,"abstract":"Let $X$ be a Banach space, $Ain B(X)$ and $M$ be an invariant subspace of $A$. We present an alternative proof that, if the spectrum of the restriction of $A$ to $M$ contains a point that is in any given hole in the spectrum of $A$, then the entire hole is in the spectrum of the restriction.","PeriodicalId":8426,"journal":{"name":"arXiv: Functional Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74011430","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 : 2020-04-01DOI: 10.15330/cmp.12.1.107-110
M. A. Mytrofanov, A. Ravsky
Let $X$ be a real separable normed space $X$ admitting a separating polynomial. We prove that each continuous function from a subset $A$ of $X$ to a real Banach space can be uniformly approximated by restrictions to $A$ of functions which are analytic on open subsets of $X$. Also we prove that each continuous function to a complex Banach space from a complex separable normed space admitting a separating $*$-polynomial can be uniformly approximated by $*$-analytic functions.
设X$是一个实可分离赋范空间X$，允许一个分离多项式。证明了从$X$的子集$ a $到实巴拿赫空间的每一个连续函数都可以由$X$的开子集上解析函数的$ a $的限制一致逼近。同时证明了从具有分离的$*$-多项式的复可分赋范空间到复巴拿赫空间的每一个连续函数都可以用$*$-解析函数一致逼近。
{"title":"A note on approximation of continuous functions on normed spaces","authors":"M. A. Mytrofanov, A. Ravsky","doi":"10.15330/cmp.12.1.107-110","DOIUrl":"https://doi.org/10.15330/cmp.12.1.107-110","url":null,"abstract":"Let $X$ be a real separable normed space $X$ admitting a separating polynomial. We prove that each continuous function from a subset $A$ of $X$ to a real Banach space can be uniformly approximated by restrictions to $A$ of functions which are analytic on open subsets of $X$. Also we prove that each continuous function to a complex Banach space from a complex separable normed space admitting a separating $*$-polynomial can be uniformly approximated by $*$-analytic functions.","PeriodicalId":8426,"journal":{"name":"arXiv: Functional Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85229069","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 the multiplier algebras $A(mathcal{H})$ obtained as the closure of the polynomials on certain reproducing kernel Hilbert spaces $mathcal{H}$ on the ball $mathbb{B}_d$ of $mathbb{C}^d$. Our results apply, in particular, to the Drury-Arveson space, the Dirichlet space and the Hardy space on the ball. We first obtain a complete description of the dual and second dual spaces of $A(mathcal H)$ in terms of the complementary bands of Henkin and totally singular measures for $operatorname{Mult}(mathcal{H})$. This is applied to obtain several definitive results in interpolation. In particular, we establish a sharp peak interpolation result for compact $operatorname{Mult}(mathcal{H})$-totally null sets as well as a Pick and peak interpolation theorem. Conversely, we show that a mere interpolation set is $operatorname{Mult}(mathcal{H})$-totally null.
研究了在$mathbb{C}^d$的$mathbb{B}_d$上的若干可再生核希尔伯特空间$mathcal{H}$上多项式的闭包所得到的乘子代数$A(mathcal{H})$。我们的结果特别适用于球上的Drury-Arveson空间、Dirichlet空间和Hardy空间。首先给出了$ a (mathcal H)$的对偶空间和第二对偶空间在$operatorname{Mult}(mathcal{H})$的Henkin互补带和全奇异测度的完备描述。该方法在插值中得到了几个明确的结果。特别地，我们建立了紧$operatorname{Mult}(mathcal{H})$-全空集的尖峰插值结果以及Pick和peak插值定理。相反，我们证明了一个单纯的插值集$operatorname{Mult}(mathcal{H})$-完全为空。
{"title":"Interpolation and duality in algebras of multipliers on the ball","authors":"K. Davidson, Michael Hartz","doi":"10.4171/jems/1245","DOIUrl":"https://doi.org/10.4171/jems/1245","url":null,"abstract":"We study the multiplier algebras $A(mathcal{H})$ obtained as the closure of the polynomials on certain reproducing kernel Hilbert spaces $mathcal{H}$ on the ball $mathbb{B}_d$ of $mathbb{C}^d$. Our results apply, in particular, to the Drury-Arveson space, the Dirichlet space and the Hardy space on the ball. We first obtain a complete description of the dual and second dual spaces of $A(mathcal H)$ in terms of the complementary bands of Henkin and totally singular measures for $operatorname{Mult}(mathcal{H})$. This is applied to obtain several definitive results in interpolation. In particular, we establish a sharp peak interpolation result for compact $operatorname{Mult}(mathcal{H})$-totally null sets as well as a Pick and peak interpolation theorem. Conversely, we show that a mere interpolation set is $operatorname{Mult}(mathcal{H})$-totally null.","PeriodicalId":8426,"journal":{"name":"arXiv: Functional Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76882106","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}
It is well known that the set of all $ n times n $ matrices with distinct eigenvalues is a dense subset of the set of all real or complex $ n times n $ matrices. In [Hartfiel, D. J. Dense sets of diagonalizable matrices. Proc. Amer. Math. Soc., 123(6): 1669-1672, 1995.], the author established a necessary and sufficient condition for a subspace of the set of all $n times n$ matrices to have a dense subset of matrices with distinct eigenvalues. We are interested in finding a few necessary and sufficient conditions for a subset of the set of all $n times n$ real or complex matrices to have a dense subset of matrices with distinct eigenvalues. Some of our results are generalizing the results of Hartfiel. Also, we study the existence of dense subsets of matrices with distinct singular values, distinct analytic eigenvalues, and distinct analytic singular values, respectively, in the subsets of the set of all real or complex matrices.
众所周知，所有具有不同特征值的$ n 乘以n $矩阵的集合是所有实或复$ n 乘以n $矩阵的集合的密集子集。在[hartfield, D. j]中，可对角化矩阵的密集集。Proc,阿米尔。数学。Soc。中国生物医学工程学报，32(6):1669-1672,1995。]，建立了所有$n × n$矩阵集合的子空间具有具有不同特征值的矩阵的稠密子集的充分必要条件。我们感兴趣的是找到一些必要和充分条件，使得所有n × n的实矩阵或复矩阵的集合的子集有一个具有不同特征值的矩阵的稠密子集。我们的一些结果推广了哈特菲尔的结果。此外，我们还研究了在所有实矩阵或复矩阵集合的子集中，具有不同奇异值矩阵、不同解析特征值矩阵和不同解析奇异值矩阵的密集子集的存在性。
{"title":"On the dense subsets of matrices with distinct eigenvalues and distinct singular values","authors":"Himadri Lal Das, M. Kannan","doi":"10.13001/ela.2020.5329","DOIUrl":"https://doi.org/10.13001/ela.2020.5329","url":null,"abstract":"It is well known that the set of all $ n times n $ matrices with distinct eigenvalues is a dense subset of the set of all real or complex $ n times n $ matrices. In [Hartfiel, D. J. Dense sets of diagonalizable matrices. Proc. Amer. Math. Soc., 123(6): 1669-1672, 1995.], the author established a necessary and sufficient condition for a subspace of the set of all $n times n$ matrices to have a dense subset of matrices with distinct eigenvalues. We are interested in finding a few necessary and sufficient conditions for a subset of the set of all $n times n$ real or complex matrices to have a dense subset of matrices with distinct eigenvalues. Some of our results are generalizing the results of Hartfiel. Also, we study the existence of dense subsets of matrices with distinct singular values, distinct analytic eigenvalues, and distinct analytic singular values, respectively, in the subsets of the set of all real or complex matrices.","PeriodicalId":8426,"journal":{"name":"arXiv: Functional Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-03-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84973630","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 : 2020-03-24DOI: 10.22541/AU.159991025.58949527
M. Sababheh, Shigeru Furuichi, H. Moradi
Convex functions have played a major role in the field of Mathematical inequalities. In this paper, we introduce a new concept related to convexity, which proves better estimates when the function is somehow more convex than another. In particular, we define what we called $g-$convexity as a generalization of $log-$convexity. Then we prove that $g-$convex functions have better estimates in certain known inequalities like the Hermite-Hadard inequality, super additivity of convex functions, the Majorization inequality and some means inequalities. Strongly related to this, we define the index of convexity as a measure of ``how much the function is convex". Applications including Hilbert space operators, matrices and entropies will be presented in the end.
{"title":"A new treatment of convex functions","authors":"M. Sababheh, Shigeru Furuichi, H. Moradi","doi":"10.22541/AU.159991025.58949527","DOIUrl":"https://doi.org/10.22541/AU.159991025.58949527","url":null,"abstract":"Convex functions have played a major role in the field of Mathematical inequalities. In this paper, we introduce a new concept related to convexity, which proves better estimates when the function is somehow more convex than another. In particular, we define what we called $g-$convexity as a generalization of $log-$convexity. Then we prove that $g-$convex functions have better estimates in certain known inequalities like the Hermite-Hadard inequality, super additivity of convex functions, the Majorization inequality and some means inequalities. Strongly related to this, we define the index of convexity as a measure of ``how much the function is convex\". \u0000Applications including Hilbert space operators, matrices and entropies will be presented in the end.","PeriodicalId":8426,"journal":{"name":"arXiv: Functional Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-03-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89658028","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 : 2020-03-20DOI: 10.1007/978-3-030-51945-2_7
V. Didenko, B. Silbermann
{"title":"Invertibility Issues for Toeplitz Plus Hankel Operators and Their Close Relatives","authors":"V. Didenko, B. Silbermann","doi":"10.1007/978-3-030-51945-2_7","DOIUrl":"https://doi.org/10.1007/978-3-030-51945-2_7","url":null,"abstract":"","PeriodicalId":8426,"journal":{"name":"arXiv: Functional Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88302385","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}
It is common that a Sobolev space defined on $mathbb{R}^m$ has a non-compact embedding into an $L^p$-space, but it has subspaces for which this embedding becomes compact. There are three well known cases of such subspaces, the Rellich compactness, for a subspace of functions on a bounded domain (or an unbounded domain, sufficiently thin at infinity), the Strauss compactness, for a subspace of radially symmetric functions in $mathbb{R}^m$, and the weighted Sobolev spaces. Known generalizations of Strauss compactness include subspaces of functions with block-radial symmetry, subspaces of functions with certain symmetries on Riemannian manifolds, as well as similar subspaces of more general Besov and Triebel-Lizorkin spaces. Presence of symmetries can be interpreted in terms of the rising critical Sobolev exponent corresponding to the smaller effective dimension of the quotient space.
{"title":"On compact subsets of Sobolev spaces on manifolds","authors":"L. Skrzypczak, C. Tintarev","doi":"10.1090/tran/8322","DOIUrl":"https://doi.org/10.1090/tran/8322","url":null,"abstract":"It is common that a Sobolev space defined on $mathbb{R}^m$ has a non-compact embedding into an $L^p$-space, but it has subspaces for which this embedding becomes compact. There are three well known cases of such subspaces, the Rellich compactness, for a subspace of functions on a bounded domain (or an unbounded domain, sufficiently thin at infinity), the Strauss compactness, for a subspace of radially symmetric functions in $mathbb{R}^m$, and the weighted Sobolev spaces. Known generalizations of Strauss compactness include subspaces of functions with block-radial symmetry, subspaces of functions with certain symmetries on Riemannian manifolds, as well as similar subspaces of more general Besov and Triebel-Lizorkin spaces. Presence of symmetries can be interpreted in terms of the rising critical Sobolev exponent corresponding to the smaller effective dimension of the quotient space.","PeriodicalId":8426,"journal":{"name":"arXiv: Functional Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84486046","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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