Mostrando recursos 1 - 20 de 24

  1. 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.

  2. 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.

  3. 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...

  4. 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 $\alpha_{L_{1}}(X^{*})=\gamma_{C(\Delta)}(X)=\gamma_{C[0,1]}(X)$ .

  5. 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.

  6. 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.

  7. 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.

  8. 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.

  9. Nonlinear isometries between function spaces

    Roberts, Kathleen; Lee, Kristopher
    We demonstrate that any surjective isometry $T\colon \mathcal{A}\to \mathcal{B}$ not assumed to be linear between unital, completely regular subspaces of complex-valued, continuous functions on compact Hausdorff spaces is of the form \begin{equation*}T(f)=T(0)+\operatorname{Re}[\mu \cdot(f\circ\tau)]+i\operatorname{Im}[\nu \cdot(f\circ\rho)],\end{equation*} where $\mu$ and $\nu$ are continuous and unimodular, there exists a clopen set $K$ with $\nu=\mu$ on $K$ and $\nu=-\mu$ on $K^{c}$ , and $\tau$ and $\rho$ are homeomorphisms.

  10. Chlodowsky–Szasz–Appell-type operators for functions of two variables

    Sidharth, Manjari; Acu, Ana Maria; Agrawal, P. N.
    This article deals with the approximation properties of the bivariate operators which are the combination of Bernstein–Chlodowsky operators and the Szász operators involving Appell polynomials. We investigate the degree of approximation of the operators with the help of the complete modulus of continuity and the partial moduli of continuity. In the last section of the paper, we introduce the generalized Boolean sum (GBS) of these bivariate Chlodowsky–Szasz–Appell-type operators and examine the order of approximation in the Bögel space of continuous functions by means of the mixed modulus of smoothness.

  11. Equivalent results to Banach’s contraction principle

    Berzig, Maher; Rus, Cristina-Olimpia; Rus, Mircea-Dan
    We present two versions of the well-known Banach contraction principle: one in the context of extended metric spaces for which the distance mapping is allowed to be infinite, the other in the context of metric spaces endowed with a compatible binary relation. We also point out that these two results and the Banach contraction principle are actually equivalent.

  12. Sherman type theorem on $C^{\ast}$ -algebras

    Niezgoda, Marek
    In this paper, a new definition of majorization for $C^{\ast}$ -algebras is introduced. Sherman’s inequality is extended to self-adjoint operators and positive linear maps by applying the method of premajorization used for comparing two tuples of objects. A general result in a matrix setting is established. Special cases of the main theorem are studied. In particular, a HLPK-type inequality is derived.

  13. Local Lie derivations on certain operator algebras

    Liu, Dan; Zhang, Jianhua
    In this paper, we investigate local Lie derivations of a certain class of operator algebras and show that, under certain conditions, every local Lie derivation of such an algebra is a Lie derivation.

  14. Hadamard gap series in weighted-type spaces on the unit ball

    Hu, Bingyang; Li, Songxiao
    We give a sufficient and necessary condition for an analytic function $f(z)$ on the unit ball $\mathbb{B}$ in $\mathbb{C}^{n}$ with Hadamard gaps, that is, for $f(z)=\sum_{k=1}^{\infty}P_{n_{k}}(z)$ where $P_{n_{k}}(z)$ is a homogeneous polynomial of degree $n_{k}$ and $n_{k+1}/n_{k}\ge c\gt 1$ for all $k\in\mathbb{N}$ , to belong to the weighted-type space $H^{\infty}_{\mu}$ and the corresponding little weighted-type space $H^{\infty}_{\mu,0}$ under some condition posed on the weighted funtion $\mu$ . We also study the growth rate of those functions in $H^{\infty}_{\mu}$ .

  15. Triple solutions for quasilinear one-dimensional $p$ -Laplacian elliptic equations in the whole space

    Bonanno, Gabriele; O’Regan, Donal; Vetro, Francesca
    In this paper, we establish the existence of three possibly nontrivial solutions for a Dirichlet problem on the real line without assuming on the nonlinearity asymptotic conditions at infinity. As a particular case, when the nonlinearity is superlinear at zero and sublinear at infinity, the existence of two nontrivial solutions is obtained. This approach is based on variational methods and, more precisely, a critical points theorem, which assumes a more general condition than the classical Palais–Smale condition, is exploited.

  16. Some operator inequalities for unitarily invariant norms

    Zhao, Jianguo; Wu, Junliang
    This note aims to present some operator inequalities for unitarily invariant norms. First, a Zhan-type inequality for unitarily invariant norms is given. Moreover, some operator inequalities for the Cauchy–Schwarz type are also established.

  17. $L^{p}$ -Inequalities and Parseval-type relations for the Mehler–Fock transform of general order

    González, Benito J.; Negrín, Emilio R.
    In this article we study new $L^{p}$ -boundedness properties for the Mehler–Fock transform of general order on the spaces $L^{p}((0,\infty),e^{\alpha x}dx)$ and $L^{p}((0,\infty),(1+x)^{\gamma}dx)$ , $1\leq p\leq\infty$ , and $\alpha,\gamma\in\mathbb{R}$ . We also obtain Parseval-type relations over these spaces.

  18. Geometric constants of $\pi/2$ -rotation invariant norms on $\mathbb{R}^{2}$

    Tomizawa, Yukino; Mitani, Ken-Ichi; Saito, Kichi-Suke; Tanaka, Ryotaro
    In this article, we study the (modified) von Neumann–Jordan constant and Zbăganu constant of $\pi/2$ -rotation invariant norms on $\mathbb{R}^{2}$ . Some estimations of these geometric constants are given. As an application, we construct various examples consisting of $\pi/2$ -rotation invariant norms.

  19. Nonsimplicity of certain universal ${\mathrm{C}}^{\ast}$ -algebras

    de Jeu, Marcel; El Harti, Rachid; Pinto, Paulo R.
    Given $n\geq2$ , $z_{ij}\in\mathbb{T}$ such that $z_{ij}=\overline{z}_{ji}$ for $1\leq i,j\leq n$ and $z_{ii}=1$ for $1\leq i\leq n$ , and integers $p_{1},\ldots,p_{n}\geq1$ , we show that the universal ${\mathrm{C}}^{\ast}$ -algebra generated by unitaries $u_{1},\ldots,u_{n}$ such that $u_{i}^{p_{i}}u_{j}^{p_{j}}=z_{ij}u_{j}^{p_{j}}u_{i}^{p_{i}}$ for $1\leq i,j\leq n$ is not simple if at least one exponent $p_{i}$ is at least two. We indicate how the method of proof by “working with various quotients” can be used to establish nonsimplicity of universal ${\mathrm{C}}^{\ast}$ -algebras in other cases.

  20. Weighted backward shift operators with invariant distributionally scrambled subsets

    Wu, Xinxing; Wang, Lidong; Chen, Guanrong
    We obtain a sufficient condition to ensure that weighted backward shift operators on Köthe sequence spaces $\lambda_{p}(A)$ admit an invariant distributionally $\varepsilon$ -scrambled subset for any $0\lt \varepsilon\lt \operatorname{diam}\lambda_{p}(A)$ . In particular, every Devaney chaotic weighted backward shift operator on $\lambda_{p}(A)$ supports such a subset.

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