Mostrando recursos 1 - 20 de 48

  1. Reflexive sets of operators

    Bračič, Janko; Diogo, Cristina; Zajac, Michal
    For a set $\mathcal{M}$ of operators on a complex Banach space $\mathscr{X}$ , the reflexive cover of $\mathcal{M}$ is the set $\operatorname{Ref}(\mathcal{M})$ of all those operators $T$ satisfying $Tx\in\overline{\mathcal{M}x}$ for every $x\in\mathscr{X}$ . Set $\mathcal{M}$ is reflexive if $\operatorname{Ref}(\mathcal{M})=\mathcal{M}$ . The notion is well known, especially for Banach algebras or closed spaces of operators, because it is related to the problem of invariant subspaces. We study reflexivity for general sets of operators. We are interested in how the reflexive cover behaves towards basic operations between sets of operators. It is easily seen that the intersection of an arbitrary family of...

  2. Disjointness-preserving orthogonally additive operators in vector lattices

    Abasov, Nariman; Pliev, Marat
    In this article, we investigate disjointness-preserving orthogonally additive operators in the setting of vector lattices. First, we present a formula for the band projection onto the band generated by a single positive, disjointness-preserving, order-bounded, orthogonally additive operator. Then we prove a Radon–Nikodým theorem for a positive, disjointness-preserving, order-bounded, orthogonally additive operator defined on a vector lattice $E$ , taking values in a Dedekind-complete vector lattice $F$ . We conclude by obtaining an analytical representation for a nonlinear lattice homomorphism between order ideals of spaces of measurable almost everywhere finite functions.

  3. Perturbation analysis of the Moore–Penrose metric generalized inverse with applications

    Cao, Jianbing; Xue, Yifeng
    In this article, based on some geometric properties of Banach spaces and one feature of the metric projection, we introduce a new class of bounded linear operators satisfying the so-called $(\alpha,\beta)$ -USU (uniformly strong uniqueness) property. This new convenient property allows us to take the study of the stability problem of the Moore–Penrose metric generalized inverse a step further. As a result, we obtain various perturbation bounds of the Moore–Penrose metric generalized inverse of the perturbed operator. They offer the advantage that we do not need the quasiadditivity assumption, and the results obtained appear to be the most general case...

  4. Square function inequalities for monotone bases in $L^{1}$

    Osękowski, Adam
    We describe a novel method of handling general sharp square function inequalities for monotone bases and contractive projections in $L^{1}$ . The technique rests on the construction of an appropriate special function enjoying certain size and convexity-type properties. As an illustration, we establish a strong $L^{1}\to L^{1}$ and a weak-type $L^{1}\to L^{1,\infty}$ estimate for square functions.

  5. Sharp weighted bounds for fractional integrals via the two-weight theory

    Kokilashvili, Vakhtang; Meskhi, Alexander; Zaighum, Muhammad Asad
    We derive sharp weighted norm estimates for positive kernel operators on spaces of homogeneous type. Similar problems are studied for one-sided fractional integrals. Bounds of weighted norms are of mixed type. The problems are studied using the two-weight theory of positive kernel operators. As special cases, we derive sharp weighted estimates in terms of Muckenhoupt characteristics.

  6. Rotation of Gaussian paths on Wiener space with applications

    Chang, Seung Jun; Choi, Jae Gil
    In this paper we first develop the rotation theorem of the Gaussian paths on Wiener space. We next analyze the generalized analytic Fourier–Feynman transform. As an application of our rotation theorem, we represent the multiple generalized analytic Fourier–Feynman transform as a single generalized Fourier–Feynman transform.

  7. Pointwise entangled ergodic theorems for Dunford–Schwartz operators

    Kunszenti-Kovács, Dávid
    We investigate pointwise convergence of entangled ergodic averages of Dunford–Schwartz operators $T_{0},T_{1},\ldots,T_{m}$ on a Borel probability space. These averages take the form ¶ \[\frac{1}{N^{k}}\sum_{1\leq n_{1},\ldots,n_{k}\leq N}T_{m}^{n_{\alpha(m)}}A_{m-1}T^{n_{\alpha(m-1)}}_{m-1}\cdots A_{2}T_{2}^{n_{\alpha(2)}}A_{1}T_{1}^{n_{\alpha(1)}}f,\] where $f\in L^{p}(X,\mu)$ for some $1\leq p\lt \infty$ , and $\alpha:\{1,\ldots,m\}\to\{1,\ldots,k\}$ encodes the entanglement. We prove that, under some joint boundedness and twisted compactness conditions on the pairs $(A_{i},T_{i})$ , convergence holds almost everywhere for all $f\in L^{p}$ . We also present an extension to polynomial powers in the case $p=2$ , in addition to a continuous version concerning Dunford–Schwartz $C_{0}$ -semigroups.

  8. A generalized Schur complement for nonnegative operators on linear spaces

    Friedrich, J.; Günther, M.; Klotz, L.
    Extending the corresponding notion for matrices or bounded linear operators on a Hilbert space, we define a generalized Schur complement for a nonnegative linear operator mapping a linear space into its dual, and we derive some of its properties.

  9. On a generalized uniform zero-two law for positive contractions of noncommutative $L_{1}$ -spaces and its vector-valued extension

    Ganiev, Inomjon; Mukhamedov, Farrukh; Bekbaev, Dilmurod
    Ornstein and Sucheston first proved that for a given positive contraction $T:L_{1}\to L_{1}$ there exists $m\in{\mathbb{N}}$ such that if $\Vert T^{m+1}-T^{m}\Vert \lt 2$ , then $\lim_{n\to\infty}\Vert T^{n+1}-T^{n}\Vert =0$ . This result was referred to as the zero-two law. In the present article, we prove a generalized uniform zero-two law for the multiparametric family of positive contractions of noncommutative $L_{1}$ -spaces. Moreover, we also establish a vector-valued analogue of the uniform zero-two law for positive contractions of $L_{1}(M,\Phi)$ —the noncommutative $L_{1}$ -spaces associated with center-valued traces.

  10. Cohomology for small categories: $k$ -graphs and groupoids

    Gillaspy, Elizabeth; Kumjian, Alexander
    Given a higher-rank graph $\Lambda$ , we investigate the relationship between the cohomology of $\Lambda$ and the cohomology of the associated groupoid $\mathcal{G}_{\Lambda}$ . We define an exact functor between the Abelian category of right modules over a higher-rank graph $\Lambda$ and the category of $\mathcal{G}_{\Lambda}$ -sheaves, where $\mathcal{G}_{\Lambda}$ is the path groupoid of $\Lambda$ . We use this functor to construct compatible homomorphisms from both the cohomology of $\Lambda$ with coefficients in a right $\Lambda$ -module, and the continuous cocycle cohomology of $\mathcal{G}_{\Lambda}$ with values in the corresponding $\mathcal{G}_{\Lambda}$ -sheaf, into the sheaf cohomology of $\mathcal{G}_{\Lambda}$ .

  11. Complex interpolation of predual spaces of general local Morrey-type spaces

    Hakim, Denny Ivanal
    In this article, we investigate the complex interpolation of predual spaces of general local Morrey-type spaces. By showing that these spaces are equal to the associate space of general local Morrey-type spaces, we prove that predual spaces of general local Morrey-type spaces behave well under the first complex interpolation.

  12. Kolmogorov-type and general extension results for nonlinear expectations

    Denk, Robert; Kupper, Michael; Nendel, Max
    We provide extension procedures for nonlinear expectations to the space of all bounded measurable functions. We first discuss a maximal extension for convex expectations which have a representation in terms of finitely additive measures. One of the main results of this article is an extension procedure for convex expectations which are continuous from above and therefore admit a representation in terms of countably additive measures. This can be seen as a nonlinear version of the Daniell–Stone theorem. From this, we deduce a robust Kolmogorov extension theorem which is then used to extend nonlinear kernels to an infinite-dimensional path space. We...

  13. 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...

  14. 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.

  15. 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 $\mathcal{V}^{*}(\varphi)$ .

  16. 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.

  17. 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.

  18. 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.

  19. 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...

  20. 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 $C_{0}$...

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