Mostrando recursos 1 - 20 de 36

  1. On an approximation of $2$ -dimensional Walsh–Fourier series in martingale Hardy spaces

    Persson, L. E.; Tephnadze, G.; Wall, P.
    In this paper, we investigate convergence and divergence of partial sums with respect to the $2$ -dimensional Walsh system on the martingale Hardy spaces. In particular, we find some conditions for the modulus of continuity which provide convergence of partial sums of Walsh–Fourier series. We also show that these conditions are in a sense necessary and sufficient.

  2. On the $p$ -Schur property of Banach spaces

    Dehghani, Mohammad B.; Moshtaghioun, S. Mohammad
    We introduce the notion of the $p$ -Schur property ( $1\leq p\leq\infty$ ) as a generalization of the Schur property of Banach spaces, and then we present a number of basic properties and some examples. We also study its relation with some geometric properties of Banach spaces, such as the Gelfand–Phillips property. Moreover, we verify some necessary and sufficient conditions for the $p$ -Schur property of some closed subspaces of operator spaces.

  3. On solving proximal split feasibility problems and applications

    Witthayarat, Uamporn; Cho, Yeol Je; Cholamjiak, Prasit
    We study the problem of proximal split feasibility of two objective convex functions in Hilbert spaces. We prove that, under suitable conditions, certain strong convergence theorems of the Halpern-type algorithm present solutions to the proximal split feasibility problem. Finally, we provide some related applications as well as numerical experiments.

  4. On the modulus of disjointness-preserving operators and $b$ - $AM$ -compact operators on Banach lattices

    Haghnezhad Azar, Kazem; Alavizadeh, Razi
    We study several properties of the modulus of order bounded disjointness-preserving operators. We show that, if $T$ is an order bounded disjointness-preserving operator, then $T$ and $\vert T\vert $ have the same compactness property for several types of compactness. Finally, we characterize Banach lattices having $b$ - $\mathit{AM}$ -compact (resp., $\mathit{AM}$ -compact) operators defined between them as having a modulus that is $b$ - $\mathit{AM}$ -compact (resp., $\mathit{AM}$ -compact).

  5. Atomic decomposition of variable Hardy spaces via Littlewood–Paley–Stein theory

    Tan, Jian
    The purpose of this paper is to give a new atomic decomposition for variable Hardy spaces via the discrete Littlewood–Paley–Stein theory. As an application of this decomposition, we assume that $T$ is a linear operator bounded on $L^{q}$ and $H^{p(\cdot)}$ , and we thus obtain that $T$ can be extended to a bounded operator from $H^{p(\cdot)}$ to $L^{p(\cdot)}$ .

  6. Inhomogeneous Lipschitz spaces of variable order and their applications

    Tan, Jian; Zhao, Jiman
    In this article, the authors first give a Littlewood–Paley characterization for inhomogeneous Lipschitz spaces of variable order with the help of inhomogeneous Calderón identity and almost-orthogonality estimates. As applications, the boundedness of inhomogeneous Calderón–Zygmund singular integral operators of order $(\epsilon,\sigma)$ on these spaces has been presented. Finally, we note that a class of pseudodifferential operators $T_{a}\in\mathcal{O}pS_{1,1}^{0}$ are continuous on the inhomogeneous Lipschitz spaces of variable order as a corollary. We may observe that those operators are not, in general, continuous in $L^{2}$ .

  7. Bases in some spaces of Whitney functions

    Goncharov, Alexander; Ural, Zeliha
    We construct topological bases in spaces of Whitney functions on Cantor sets, which were introduced by the first author. By means of suitable individual extensions of basis elements, we construct a linear continuous extension operator, when it exists for the corresponding space. In general, elements of the basis are restrictions of polynomials to certain subsets. In the case of small sets, we can present strict polynomial bases as well.

  8. A treatment of strongly operator-convex functions that does not require any knowledge of operator algebras

    Brown, Lawrence G.
    In a previous article, we proved the equivalence of six conditions on a continuous function $f$ on an interval. These conditions determine a subset of the set of operator-convex functions whose elements are called strongly operator-convex. Two of the six conditions involve operator-algebraic semicontinuity theory, as given by Akemann and Pedersen, and the other four conditions do not involve operator algebras at all. Two of these conditions are operator inequalities, one is a global condition on $f$ , and the fourth is an integral representation of $f$ , stronger than the usual integral representation for operator-convex functions. The purpose of...

  9. Scattered locally $C^{\ast}$ -algebras

    Joiţa, Maria
    In this article, we introduce the notion of a scattered locally $C^{\ast}$ -algebra and we give the conditions for a locally $C^{\ast}$ -algebra to be scattered. Given an action $\alpha$ of a locally compact group $G$ on a scattered locally $C^{\ast}$ -algebra $A[\tau_{\Gamma}]$ , it is natural to ask under what conditions the crossed product $A[\tau_{\Gamma}]\times_{\alpha}G$ is also scattered. We obtain some results concerning this question.

  10. Perturbation bounds for the Moore–Penrose metric generalized inverse in some Banach spaces

    Cao, Jianbing; Zhang, Wanqin
    Let $X,Y$ be Banach spaces, and let $T$ , $\delta T:X\to Y$ be bounded linear operators. Put $\bar{T}=T+\delta T$ . In this article, utilizing the gap between closed subspaces and the perturbation bounds of metric projections, we first present some error estimates of the upper bound of $\Vert \bar{T}^{M}-T^{M}\Vert $ in $L^{p}$ ( $1\lt p\lt +\infty$ ) spaces. Then, by using the concept of strong uniqueness and modulus of convexity, we further investigate the corresponding perturbation bound $\Vert \bar{T}^{M}-T^{M}\Vert $ in uniformly convex Banach spaces.

  11. A new algorithm for the symmetric solution of the matrix equations $AXB=E$ and $CXD=F$

    Li, Chunmei; Duan, Xuefeng; Li, Juan; Yu, Sitting
    We propose a new iterative algorithm to compute the symmetric solution of the matrix equations $AXB=E$ and $CXD=F$ . The greatest advantage of this new algorithm is higher speed and lower computational cost at each step compared with existing numerical algorithms. We state the solutions of these matrix equations as the intersection point of some closed convex sets, and then we use the alternating projection method to solve them. Finally, we use some numerical examples to show that the new algorithm is feasible and effective.

  12. Convergence properties of nets of operators

    Nasaireh, Fadel; Popa, Dorian; Rasa, Ioan
    We consider nets $(T_{j})$ of operators acting on complex functions, and we investigate the algebraic and the topological structure of the set $\{f:T_{j}(|f|^{2})-|T_{j}f|^{2}\rightarrow 0\}$ . Our results extend and improve some known results from the literature, which are connected with Korovkin’s theorem. Applications to Abel–Poisson-type operators and Bernstein-type operators are given.

  13. Involutions in algebras related to second duals of hypergroup algebras

    Medghalchi, Alireza; Ramezani, Ramin
    Let $K$ be a hypergroup. The purpose of this article is to study the question of involutions on algebras $M(K)^{**}$ , $L(K)^{**}$ , and $L_{c}(K)^{**}$ . We show that the natural involution of $M(K)$ has the canonical extension to $M(K)^{**}$ if and only if the natural involution of $L(K)$ has the canonical extension to $L(K)^{**}$ . Also, we give necessary and sufficient conditions for $M(K)^{**}$ and $L(K)^{**}$ to admit an involution extending the natural involution of $M(K)$ when $K$ is left amenable. Finally, we find the necessary and sufficient conditions for $L_{c}(K)^{**}$ to admit an involution.

  14. Minimal reducing subspaces of an operator-weighted shift

    Hazarika, Munmun; Gogoi, Pearl S.
    We introduce a family $\mathcal{T}$ consisting of invertible matrices with exactly one nonzero entry in each row and each column. The elements of $\mathcal{T}$ are possibly mutually noncommuting, and they need not be normal or self-adjoint. We consider an operator-valued unilateral weighted shift $W$ with a uniformly bounded sequence of weights belonging to $\mathcal{T}$ , and we describe its minimal reducing subspaces.

  15. Supporting vectors of continuous linear operators

    Cobos-Sánchez, Clemente; García-Pacheco, Francisco Javier; Moreno-Pulido, Soledad; Sáez-Martínez, Sol
    The set of supporting vectors of a continuous linear operator, that is, the normalized vectors at which the operator attains its norm, is decomposed into its convex components. In the complex case, the set of supporting vectors of a nonzero functional is proved to be path-connected. We also introduce the concept of generalized supporting vectors for a sequence of operators as the normalized vectors that maximize the summation of the squared norm of those operators. We determine the set of generalized supporting vectors for the particular case of a finite sequence of real matrices. Finally, we unveil the relation between...

  16. A quantitative version of the Johnson–Rosenthal theorem

    Chen, Dongyang
    Let $X,Y$ be Banach spaces. We define \begin{equation*}\alpha_{Y}(X)=\sup\{\vert T^{-1}\vert^{-1}:T:Y\rightarrow X\mbox{ is an isomorphism with }\vert T\vert \leq1\}.\end{equation*} If there is no isomorphism from $Y$ to $X$ , we set $\alpha_{Y}(X)=0$ , and ¶ \begin{equation*}\gamma_{Y}(X)=\sup\{\delta(T):T:X\rightarrow Y\mbox{ is asurjective operator with }\vert T\vert \leq1\},\end{equation*} where $\delta(T)=\sup\{\delta\gt 0:\delta B_{Y}\subseteq TB_{X}\}$ . If there is no surjective operator from $X$ onto $Y$ , we set $\gamma_{Y}(X)=0$ . We prove that for a separable space $X$ , $\alpha_{l_{1}}(X^{*})=\gamma_{c_{0}}(X)$ and $\alpha_{L_{1}}(X^{*})=\gamma_{C(\Delta)}(X)=\gamma_{C[0,1]}(X)$ .

  17. The weak Haagerup property for $C^{*}$ -algebras

    Meng, Qing
    We define and study the weak Haagerup property for $C^{*}$ -algebras in this article. A $C^{*}$ -algebra with the Haagerup property always has the weak Haagerup property. We prove that a discrete group has the weak Haagerup property if and only if its reduced group $C^{*}$ -algebra also has that property. Moreover, we consider the permanence of the weak Haagerup property under a few canonical constructions of $C^{*}$ -algebras.

  18. Orthogonal-preserving and surjective cubic stochastic operators

    Mukhamedov, Farrukh; Fadillah Embong, Ahmad; Rosli, Azizi
    In the present paper, we consider cubic stochastic operators, and prove that the surjectivity of such operators is equivalent to their orthogonal-preserving property. In the last section we provide a full description of orthogonal-preserving (respectively, surjective) cubic stochastic operators on the $2$ -dimensional simplex.

  19. On Fredholm completions of partial operator matrices

    Hai, Guojun; Zhang, Nan
    The aim of this article is to study the Fredholm completion problem of two-by-two partial operator matrices in which the lower-left entry is unspecified and others are specified. By using the methods of operator matrix representation and operator equation, we obtain necessity and sufficiency conditions for the partial operator matrices to have a Fredholm completion with the property that the lower-right entry of its Fredholm inverses is specified.

  20. Semifinite tracial subalgebras

    Bekjan, Turdebek N.; Oshanova, Azhar
    Let ${\mathcal{M}}$ be a semifinite von Neumann algebra, and let ${\mathcal{A}}$ be a tracial subalgebra of $\mathcal{M}$ . We show that ${\mathcal{A}}$ is a subdiagonal algebra of ${\mathcal{M}}$ if and only if it has the unique normal state extension property and is a $\tau$ -maximal tracial subalgebra, which is also equivalent to ${\mathcal{A}}$ having the unique normal state extension property and satisfying $L_{2}$ -density.

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