Mostrando recursos 1 - 20 de 48

  1. Equivalent properties of a Hilbert-type integral inequality with the best constant factor related to the Hurwitz zeta function

    Rassias, Michael; Yang, Bicheng
    By the use of methods of real analysis and weight functions, we study the equivalent properties of a Hilbert-type integral inequality with the nonhomogeneous kernel. The constant factor related to the Hurwitz zeta function is proved to be the best possible. As a corollary, a few equivalent conditions of a Hilbert-type integral inequality with the homogeneous kernel are deduced. We also consider their operator expressions.

  2. Nonexpansive bijections between unit balls of Banach spaces

    Zavarzina, Olesia
    It is known that if $M$ is a finite-dimensional Banach space, or a strictly convex space, or the space $\ell_{1}$ , then every nonexpansive bijection $F\colon B_{M}\to B_{M}$ of its unit ball $B_{M}$ is an isometry. We extend these results to nonexpansive bijections $F\colon B_{E}\to B_{M}$ between unit balls of two different Banach spaces. Namely, if $E$ is an arbitrary Banach space and $M$ is finite-dimensional or strictly convex, or the space $\ell_{1}$ , then every nonexpansive bijection $F\colon B_{E}\to B_{M}$ is an isometry.

  3. Compact and “compact” operators on standard Hilbert modules over $W^{*}$ -algebras

    Kečkić, Dragoljub J.; Lazović, Zlatko
    We construct a topology on the standard Hilbert module $l^{2}(\mathcal{A})$ over a unital $W^{*}$ -algebra $\mathcal{A}$ such that any “compact” operator (i.e., any operator in the norm closure of the linear span of the operators of the form $x\mapsto z\langle y,x\rangle$ ) maps bounded sets into totally bounded sets.

  4. Spectral properties of the Lau product $\mathcal{A}\times_{\theta}\mathcal{B}$ of Banach algebras

    Dabhi, Prakash A.; Patel, Savan K.
    Let $\mathcal{A}$ and $\mathcal{B}$ be commutative Banach algebras. Then a multiplicative linear functional $\theta$ on $\mathcal{B}$ induces a multiplication on the Cartesian product space $\mathcal{A}\times\mathcal{B}$ given by $(a,b)(c,d)=(ac+\theta(d)a+\theta(b)c,bd)$ for all $(a,b),(c,d)\in\mathcal{A}\times\mathcal{B}$ . We show that this Lau product is stable with respect to the spectral properties like the unique uniform norm property, the spectral extension property, the multiplicative Hahn–Banach property, and the unique semisimple norm property under certain conditions on $\theta$ .

  5. On operators with closed numerical ranges

    Ji, Youqing; Liang, Bin
    In this article we investigate the numerical ranges of several classes of operators. It is shown that, if we let $T$ be a hyponormal operator and let $\varepsilon\gt 0$ , then there exists a compact operator $K$ with norm less than $\varepsilon$ such that $T+K$ is hyponormal and has a closed numerical range. Moreover we prove that the statement of the above type holds for other operator classes, including weighted shifts, normaloid operators, triangular operators, and block-diagonal operators.

  6. On generalized pointwise noncyclic contractions without proximal normal structure

    Gabeleh, Moosa
    In this article, we introduce a new class of noncyclic mappings called generalized pointwise noncyclic contractions, and we prove a best proximity pair theorem for this class of noncyclic mappings in the setting of strictly convex Banach spaces. Our conclusions generalize a result due to Kirk and Royalty. We also study convergence of iterates of noncyclic contraction mappings in uniformly convex Banach spaces.

  7. Bekka-type amenabilities for unitary corepresentations of locally compact quantum groups

    Chen, Xiao
    In this short note, we further Ng’s work by extending Bekka amenability and weak Bekka amenability to general locally compact quantum groups, and we generalize some of Ng’s results to the general case. In particular, we show that a locally compact quantum group ${\mathbb{G}}$ is coamenable if and only if the contra-corepresentation of its fundamental multiplicative unitary $W_{\mathbb{G}}$ is Bekka-amenable, and that ${\mathbb{G}}$ is amenable if and only if its dual quantum group’s fundamental multiplicative unitary $W_{\widehat{\mathbb{G}}}$ is weakly Bekka-amenable.

  8. Noncommutative geometry of rational elliptic curves

    Nikolaev, Igor V.
    We study an interplay between operator algebras and the geometry of rational elliptic curves. Namely, let $\mathcal{O}_{B}$ be the Cuntz–Krieger algebra given by a square matrix $B=(b-1,1,b-2,1)$ , where $b$ is an integer greater than or equal to $2$ . We prove that there exists a dense, self-adjoint subalgebra of $\mathcal{O}_{B}$ which is isomorphic (modulo an ideal) to a twisted homogeneous coordinate ring of the rational elliptic curve $\mathcal{E}({\Bbb{Q}})=\{(x,y,z)\in{\Bbb{P}}^{2}({\Bbb{C}})\midy^{2}z=x(x-z)(x-{\frac{b-2}{b+2}}z)\}$ .

  9. Lim’s center and fixed-point theorems for isometry mappings

    Rajesh, S.; Veeramani, P.
    In this article, we prove that if $K$ is a nonempty weakly compact convex set in a Banach space such that $K$ has the hereditary fixed-point property (FPP) and $\mathfrak{F}$ is a commuting family of isometry mappings on $K$ , then there exists a point in $C(K)$ which is fixed by every member in $\mathfrak{F}$ whenever $C(K)$ is a compact set. Also, we give an example to show that $C(K)$ , the Chebyshev center of $K$ , need not be invariant under isometry maps. This example answers the question as to whether the Chebyshev center is invariant under isometry maps....

  10. Essential norm of the composition operators on the general spaces $H_{\omega,p}$ of Hardy spaces

    Rezaei, S.
    We obtain estimates for the essential norm of the composition operators acting on the general spaces $H_{\omega,p}$ of Hardy spaces. Our characterization is given in terms of generalized Nevanlinna counting functions.

  11. Convolution-continuous bilinear operators acting on Hilbert spaces of integrable functions

    Erdoğan, Ezgi; Calabuig, José M.; Sánchez Pérez, Enrique A.
    We study bilinear operators acting on a product of Hilbert spaces of integrable functions—zero-valued for couples of functions whose convolution equals zero—that we call convolution-continuous bilinear maps. We prove a factorization theorem for them, showing that they factor through $\ell^{1}$ . We also present some applications for the case when the range space has some relevant properties, such as the Orlicz or Schur properties. We prove that $\ell^{1}$ is the only Banach space for which there is a norming bilinear map which equals zero exactly in those couples of functions whose convolution is zero. We also show some examples and...

  12. Berezin transform of the absolute value of an operator

    Das, Namita; Sahoo, Madhusmita
    In this article, we concentrate on the Berezin transform of the absolute value of a bounded linear operator $T$ defined on the Bergman space $L_{a}^{2}(\mathbb{D})$ of the open unit disk. We establish some sufficient conditions on $T$ which guarantee that the Berezin transform of $|T|$ majorizes the Berezin transform of $|T^{*}|$ . We have shown that $T$ is self-adjoint and $T^{2}=T^{3}$ if and only if there exists a normal idempotent operator $S$ on $L_{a}^{2}(\mathbb{D})$ such that $\rho(T)=\rho(|S|^{2})=\rho(|S^{*}|^{2})$ , where $\rho(T)$ is the Berezin transform of $T$ . We also establish that if $T$ is compact and $|T^{n}|=|T|^{n}$ for some $n\in\mathbb{N}$...

  13. 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.

  14. 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.

  15. 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.

  16. 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).

  17. 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 $L^{p(\cdot)}$ .

  18. 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 $L^{2}$ .

  19. 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.

  20. 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...

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