Mostrando recursos 1 - 20 de 36

  1. Extrapolation theorems for $(p,q)$ -factorable operators

    Galdames-Bravo, Orlando
    The operator ideal of $(p,q)$ -factorable operators can be characterized as the class of operators that factors through the embedding $L^{q'}(\mu)\hookrightarrow L^{p}(\mu)$ for a finite measure $\mu$ , where $p,q\in[1,\infty)$ are such that $1/p+1/q\ge1$ . We prove that this operator ideal is included into a Banach operator ideal characterized by means of factorizations through $r$ th and $s$ th power factorable operators, for suitable $r,s\in[1,\infty)$ . Thus, they also factor through a positive map $L^{s}(m_{1})^{\ast}\to L^{r}(m_{2})$ , where $m_{1}$ and $m_{2}$ are vector measures. We use the properties of the spaces of $u$ -integrable functions with respect to a vector...

  2. Completely rank-nonincreasing multilinear maps

    Yousefi, Hassan
    We extend the notion of completely rank-nonincreasing (CRNI) linear maps to include the multilinear maps. We show that a bilinear map on a finite-dimensional vector space on any field is CRNI if and only if it is a skew-compression bilinear map. We also characterize CRNI continuous bilinear maps defined on the set of compact operators.

  3. Reducing subspaces for a class of nonanalytic Toeplitz operators

    Deng, Jia; Lu, Yufeng; Shi, Yanyue; Hu, Yinyin
    In this paper, we give a uniform characterization for the reducing subspaces for $T_{\varphi}$ with the symbol $\varphi(z)=z^{k}+\bar{z}^{l}$ ( $k,l\in\mathbb{Z}_{+}^{2}$ ) on the Bergman spaces over the bidisk, including the known cases that $\varphi(z_{1},z_{2})=z_{1}^{N}z_{2}^{M}$ and $\varphi(z_{1},z_{2})=z_{1}^{N}+\overline{z}_{2}^{M}$ with $N,M\in\mathbb{Z}_{+}$ . Meanwhile, the reducing subspaces for $T_{z^{N}+\overline{z}^{M}}$ ( $N,M\in \mathbb{Z}_{+}$ ) on the Bergman space over the unit disk are also described. Finally, we state these results in terms of the commutant algebra $\mathcal{V}^{*}(\varphi)$ .

  4. Toeplitz operators on weighted pluriharmonic Bergman space

    Kong, Linghui; Lu, Yufeng
    In this article, we consider some algebraic properties of Toeplitz operators on weighted pluriharmonic Bergman space on the unit ball. We characterize the commutants of Toeplitz operators whose symbols are certain separately radial functions or holomorphic monomials, and then give a partial answer to the finite-rank product problem of Toeplitz operators.

  5. Norm convergence of logarithmic means on unbounded Vilenkin groups

    Gát, György; Goginava, Ushangi
    In this paper we prove that, in the case of some unbounded Vilenkin groups, the Riesz logarithmic means converges in the norm of the spaces $X(G)$ for every $f\in X(G)$ , where by $X(G)$ we denote either the class of continuous functions with supremum norm or the class of integrable functions.

  6. New $L^{p}$ -inequalities for hyperbolic weights concerning the operators with complex Gaussian kernels

    González, Benito J.; Negrín, Emilio R.
    In this article the authors present a systematic study of several new $L^{p}$ -boundedness properties and Parseval-type relations concerning the operators with complex Gaussian kernels over the spaces $L^{p}(\mathbb{R},\cosh(\alpha x)\,dx)$ and $L^{p}(\mathbb{R},\cosh(\alpha x^{2})\,dx)$ , $1\leq p\leq\infty$ , $\alpha\in\mathbb{R}$ . Relevant connections with various earlier related results are also pointed out.

  7. A generalized Hilbert operator acting on conformally invariant spaces

    Girela, Daniel; Merchán, Noel
    If $\mu$ is a positive Borel measure on the interval $[0,1)$ , we let $\mathcal{H}_{\mu}$ be the Hankel matrix $\mathcal{H}_{\mu}=(\mu_{n,k})_{n,k\ge0}$ with entries $\mu_{n,k}=\mu_{n+k}$ , where, for $n=0,1,2,\dots$ , $\mu_{n}$ denotes the moment of order $n$ of $\mu$ . This matrix formally induces the operator ¶ \[\mathcal{H}_{\mu}(f)(z)=\sum_{n=0}^{\infty}(\sum_{k=0}^{\infty}\mu_{n,k}{a_{k}})z^{n}\] on the space of all analytic functions $f(z)=\sum_{k=0}^{\infty}a_{k}z^{k}$ , in the unit disk $\mathbb{D}$ . This is a natural generalization of the classical Hilbert operator. The action of the operators $H_{\mu}$ on Hardy spaces has been recently studied. This article is devoted to a study of the operators $H_{\mu}$ acting on certain conformally invariant spaces...

  8. Multiplicative operator functions and abstract Cauchy problems

    Früchtl, Felix
    We use the duality between functional and differential equations to solve several classes of abstract Cauchy problems related to special functions. As a general framework, we investigate operator functions which are multiplicative with respect to convolution of a hypergroup. This setting contains all representations of (hyper)groups, and properties of continuity are shown; examples are provided by translation operator functions on homogeneous Banach spaces and weakly stationary processes indexed by hypergroups. Then we show that the concept of a multiplicative operator function can be used to solve a variety of abstract Cauchy problems, containing discrete, compact, and noncompact problems, including $C_{0}$...

  9. Relatively compact sets in variable-exponent Lebesgue spaces

    Bandaliyev, Rovshan; Górka, Przemysław
    We study totally bounded sets in variable Lebesgue spaces. The full characterization of this kind of sets is given for the case of variable Lebesgue space on metric measure spaces. Furthermore, the sufficient conditions for compactness are shown without assuming $\log$ -Hölder continuity of the exponent.

  10. Lower and upper local uniform $K$ -monotonicity in symmetric spaces

    Ciesielski, Maciej
    Using the local approach to the global structure of a symmetric space $E$ , we establish a relationship between strict $K$ -monotonicity, lower (resp., upper) local uniform $K$ -monotonicity, order continuity, and the Kadec–Klee property for global convergence in measure. We also answer the question: Under which condition does upper local uniform $K$ -monotonicity coincide with upper local uniform monotonicity? Finally, we present a correlation between $K$ -order continuity and lower local uniform $K$ -monotonicity in a symmetric space $E$ under some additional assumptions on $E$ .

  11. Calderón–Lozanovskii interpolation on quasi-Banach lattices

    Raynaud, Yves; Tradacete, Pedro
    We consider the Calderón–Lozanovskii construction $\varphi(X_{0},X_{1})$ in the context of quasi-Banach lattices, and we provide an extension of a result by Ovchinnikov concerning the associated interpolation methods $\varphi^{c}$ and $\varphi^{0}$ . Our approach is based on the interpolation properties of $(\infty,1)$ -regular operators between quasi-Banach lattices.

  12. Nearly relatively compact projections in operator algebras

    Brown, Lawrence G.
    Let $A$ be a $C^{*}$ -algebra, and let $A^{**}$ be its enveloping von Neumann algebra. Akemann suggested a kind of noncommutative topology in which certain projections in $A^{**}$ play the role of open sets, and he used two operator inequalities in connection with compactness. Both of these inequalities are equivalent to compactness for a closed projection in $A^{**}$ , but only one is equivalent to relative compactness for a general projection. A third operator inequality, also related to compactness, was used by the author. The study of all three inequalities can be unified by considering a numerical invariant which is...

  13. Weighted Banach spaces of Lipschitz functions

    Jiménez-Vargas, A.
    Given a pointed metric space $X$ and a weight $v$ on $\widetilde{X}$ (the complement of the diagonal set in $X\times X$ ), let $\mathrm{Lip}_{v}(X)$ and $\mathrm{lip}_{v}(X)$ denote the Banach spaces of all scalar-valued Lipschitz functions $f$ on $X$ vanishing at the basepoint such that $v\Phi(f)$ is bounded and $v\Phi(f)$ vanishes at infinity on $\widetilde{X}$ , respectively, where $\Phi(f)$ is the de Leeuw’s map of $f$ on $\widetilde{X}$ , under the weighted Lipschitz norm. The space $\mathrm{Lip}_{v}(X)$ has an isometric predual $\mathcal{F}_{v}(X)$ and it is proved that $(\mathrm{Lip}_{v}(X),\tau_{\operatorname{bw}^{*}})=(\mathcal{F}_{v}(X)^{*},\tau_{c})$ and $\mathcal{F}_{v}(X)=((\mathrm{Lip}_{v}(X),\tau_{\operatorname{bw}^{*}})',\tau_{c})$ , where $\tau_{\operatorname{bw}^{*}}$ denotes the bounded weak∗ topology and $\tau_{c}$ the...

  14. Stability of average roughness, octahedrality, and strong diameter $2$ properties of Banach spaces with respect to absolute sums

    Haller, Rainis; Langemets, Johann; Nadel, Rihhard
    We prove that, if Banach spaces $X$ and $Y$ are $\delta$ -average rough, then their direct sum with respect to an absolute norm $N$ is $\delta/N(1,1)$ -average rough. In particular, for octahedral $X$ and $Y$ and for $p$ in $(1,\infty)$ , the space $X\oplus_{p}Y$ is $2^{1-1/p}$ -average rough, which is in general optimal. Another consequence is that for any $\delta$ in $(1,2]$ there is a Banach space which is exactly $\delta$ -average rough. We give a complete characterization when an absolute sum of two Banach spaces is octahedral or has the strong diameter 2 property. However, among all of the...

  15. Generalized frames for operators associated with atomic systems

    Li, Dongwei; Leng, Jinsong; Huang, Tingzhu
    In this paper, we investigate the g-frame and Bessel g-sequence related to a linear bounded operator $K$ in Hilbert space, which we call a $K$ -g-frame and a $K$ -dual Bessel g-sequence, respectively. Since the frame operator for a $K$ -g-frame may not be invertible, there is no classical canonical dual for a $K$ -g-frame. So we characterize the concept of a canonical $K$ -dual Bessel g-sequence of a $K$ -g-frame that generalizes the classical dual of a g-frame. Moreover, we use a family of linear operators to characterize atomic systems. We also consider the construction of new atomic systems...

  16. Vector lattices and $f$ -algebras: The classical inequalities

    Buskes, G.; Schwanke, C.
    We present some of the classical inequalities in analysis in the context of Archimedean (real or complex) vector lattices and $f$ -algebras. In particular, we prove an identity for sesquilinear maps from the Cartesian square of a vector space to a geometric mean closed Archimedean vector lattice, from which a Cauchy–Schwarz inequality follows. A reformulation of this result for sesquilinear maps with a geometric mean closed semiprime Archimedean $f$ -algebra as codomain is also given. In addition, a sufficient and necessary condition for equality is presented. We also prove a Hölder inequality for weighted geometric mean closed Archimedean $\Phi$ -algebras,...

  17. Local matrix homotopies and soft tori

    Loring, Terry A.; Vides, Fredy
    We present solutions to local connectivity problems in matrix representations of the form $C([-1,1]^{N})\to C^{*}(u_{\varepsilon},v_{\varepsilon})$ , with $C_{\varepsilon}(\mathbb{T}^{2})\twoheadrightarrow C^{*}(u_{\varepsilon},v_{\varepsilon})$ for any $\varepsilon\in[0,2]$ and any integer $n\geq1$ , where $C^{*}(u_{\varepsilon},v_{\varepsilon})\subseteq M_{n}$ is an arbitrary matrix representation of the universal $C^{*}$ -algebra $C_{\varepsilon}(\mathbb{T}^{2})$ that denotes the soft torus. We solve the connectivity problems by introducing the so-called toroidal matrix links, which can be interpreted as normal contractive matrix analogies of free homotopies in differential algebraic topology. ¶ To deal with the locality constraints, we have combined some techniques introduced in this article with some techniques from matrix geometry, combinatorial optimization, and classification...

  18. A variant of the Hankel multiplier

    Ghobber, Saifallah
    The first aim of this article is to survey and revisit some uncertainty principles for the Hankel transform by means of the Hankel multiplier. Then we define the wavelet Hankel multiplier and study its boundedness and Schatten-class properties. Finally, we prove that the wavelet Hankel multiplier is unitary equivalent to a scalar multiple of the phase space restriction operator, for which we deduce a trace formula.

  19. Approximate uniqueness for maps from $C(X)$ into simple real rank $0$ C∗-algebras

    Ng, P. W.
    Let $X$ be a finite CW-complex, and let $\mathcal{A}$ be a unital separable simple finite $\mathcal{Z}$ -stable C∗-algebra with real rank $0$ . We prove an approximate uniqueness theorem for almost multiplicative contractive completely positive linear maps from $C(X)$ into $\mathcal{A}$ . We also give conditions for when such a map can, within a certain “error,” be approximated by a finite-dimensional ∗-homomorphism.

  20. On the universal function for weighted spaces $L^{p}_{\mu}[0,1]$ , $p\geq 1$

    Grigoryan, Martin; Grigoryan, Tigran; Sargsyan, Artsrun
    In this article, we show that there exist a function $g\in L^{1}[0,1]$ and a weight function $0\lt \mu(x)\leq1$ so that $g$ is universal for each class $L^{p}_{\mu}[0,1]$ , $p\geq 1$ , with respect to signs-subseries of its Fourier–Walsh series.

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