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26. CMB 2008 (vol 51 pp. 195)

Chen, Huaihui; Gauthier, Paul
 Boundedness from Below of Composition Operators on $\alpha$-Bloch Spaces We give a necessary and sufficient condition for a composition operator on an $\alpha$-Bloch space with $\alpha\ge 1$ to be bounded below. This extends a known result for the Bloch space due to P. Ghatage, J. Yan, D. Zheng, and H. Chen. Keywords:Bloch functions, composition operatorsCategories:32A18, 30H05

27. CMB 2007 (vol 50 pp. 434)

Õzarslan, M. Ali; Duman, Oktay
 MKZ Type Operators Providing a Better Estimation on $[1/2,1)$ In the present paper, we introduce a modification of the Meyer-K\"{o}nig and Zeller (MKZ) operators which preserve the test functions $f_{0}(x)=1$ and $f_{2}(x)=x^{2}$, and we show that this modification provides a better estimation than the classical MKZ operators on the interval $[\frac{1}{2},1)$ with respect to the modulus of continuity and the Lipschitz class functionals. Furthermore, we present the $r-$th order generalization of our operators and study their approximation properties. Keywords:Meyer-KÃ¶nig and Zeller operators, Korovkin type approximation theorem, modulus of continuity, Lipschitz class functionalsCategories:41A25, 41A36

28. CMB 2005 (vol 48 pp. 505)

Bouikhalene, Belaid
 On the Generalized d'Alembert's and Wilson's Functional Equations on a Compact group Let $G$ be a compact group. Let $\sigma$ be a continuous involution of $G$. In this paper, we are concerned by the following functional equation $$\int_{G}f(xtyt^{-1})\,dt+\int_{G}f(xt\sigma(y)t^{-1})\,dt=2g(x)h(y), \quad x, y \in G,$$ where $f, g, h \colonG \mapsto \mathbb{C}$, to be determined, are complex continuous functions on $G$ such that $f$ is central. This equation generalizes d'Alembert's and Wilson's functional equations. We show that the solutions are expressed by means of characters of irreducible, continuous and unitary representations of the group $G$. Keywords:Compact groups, Functional equations, Central functions, Lie, groups, Invariant differential operators.Categories:39B32, 39B42, 22D10, 22D12, 22D15

29. CMB 2005 (vol 48 pp. 607)

Park, Efton
 Toeplitz Algebras and Extensions of\\Irrational Rotation Algebras For a given irrational number $\theta$, we define Toeplitz operators with symbols in the irrational rotation algebra ${\mathcal A}_\theta$, and we show that the $C^*$-algebra $\mathcal T({\mathcal A}_\theta)$ generated by these Toeplitz operators is an extension of ${\mathcal A}_\theta$ by the algebra of compact operators. We then use these extensions to explicitly exhibit generators of the group $KK^1({\mathcal A}_\theta,\mathbb C)$. We also prove an index theorem for $\mathcal T({\mathcal A}_\theta)$ that generalizes the standard index theorem for Toeplitz operators on the circle. Keywords:Toeplitz operators, irrational rotation algebras, index theoryCategories:47B35, 46L80

30. CMB 2005 (vol 48 pp. 175)

Borwein, David; Kratz, Werner
 Weighted Convolution Operators on $\ell_p$ The main results deal with conditions for the validity of the weighted convolution inequality $\sum_{n\in\mathbb Z}\left|b_n\sum_{k\in\mathbb Z} a_{n-k}x_k\right|^p\le C^p\sum_{k\in\mathbb Z} |x_k|^p$ when $p\ge1$. Keywords:convolution operators on $\ell_p$Categories:40G10;, 40E05

31. CMB 2004 (vol 47 pp. 615)

Randrianantoanina, Narcisse
 $C^*$-Algebras and Factorization Through Diagonal Operators Let $\cal A$ be a $C^*$-algebra and $E$ be a Banach space with the Radon-Nikodym property. We prove that if $j$ is an embedding of $E$ into an injective Banach space then for every absolutely summing operator $T:\mathcal{A}\longrightarrow E$, the composition $j \circ T$ factors through a diagonal operator from $l^{2}$ into $l^{1}$. In particular, $T$ factors through a Banach space with the Schur property. Similarly, we prove that for $2 Keywords:$C^*$-algebras, summing operators, diagonal operators,, Radon-Nikodym propertyCategories:46L50, 47D15 32. CMB 2001 (vol 44 pp. 385) Ballantine, Cristina M.  A Hypergraph with Commuting Partial Laplacians Let$F$be a totally real number field and let$\GL_{n}$be the general linear group of rank$n$over$F$. Let$\mathfrak{p}$be a prime ideal of$F$and$F_{\mathfrak{p}}$the completion of$F$with respect to the valuation induced by$\mathfrak{p}$. We will consider a finite quotient of the affine building of the group$\GL_{n}$over the field$F_{\mathfrak{p}}$. We will view this object as a hypergraph and find a set of commuting operators whose sum will be the usual adjacency operator of the graph underlying the hypergraph. Keywords:Hecke operators, buildingsCategories:11F25, 20F32 33. CMB 2000 (vol 43 pp. 406) Borwein, David  Weighted Mean Operators on$l_p$The weighted mean matrix$M_a$is the triangular matrix$\{a_k/A_n\}$, where$a_n > 0$and$A_n := a_1 + a_2 + \cdots + a_n$. It is proved that, subject to$n^c a_n$being eventually monotonic for each constant$c$and to the existence of$\alpha := \lim \frac{A_n}{na_n}$,$M_a \in B(l_p)$for$1 < p < \infty$if and only if$\alpha < p$. Keywords:weighted means, operators on$l_p$, norm estimatesCategories:47B37, 47A30, 40G05 34. CMB 2000 (vol 43 pp. 496) Xu, Yuan  Harmonic Polynomials Associated With Reflection Groups We extend Maxwell's representation of harmonic polynomials to$h$-harmonics associated to a reflection invariant weight function$h_k$. Let$\CD_i$,$1\le i \le d$, be Dunkl's operators associated with a reflection group. For any homogeneous polynomial$P$of degree$n$, we prove the polynomial$|\xb|^{2 \gamma +d-2+2n}P(\CD)\{1/|\xb|^{2 \gamma +d-2}\}$is a$h$-harmonic polynomial of degree$n$, where$\gamma = \sum k_i$and$\CD=(\CD_1,\ldots,\CD_d)$. The construction yields a basis for$h$-harmonics. We also discuss self-adjoint operators acting on the space of$h$-harmonics. Keywords:$h$-harmonics, reflection group, Dunkl's operatorsCategories:33C50, 33C45 35. CMB 1999 (vol 42 pp. 129) Baker, Andrew  Hecke Operations and the Adams$E_2$-Term Based on Elliptic Cohomology Hecke operators are used to investigate part of the$\E_2$-term of the Adams spectral sequence based on elliptic homology. The main result is a derivation of$\Ext^1$which combines use of classical Hecke operators and$p$-adic Hecke operators due to Serre. Keywords:Adams spectral sequence, elliptic cohomology, Hecke operatorsCategories:55N20, 55N22, 55T15, 11F11, 11F25 36. CMB 1998 (vol 41 pp. 129) Lee, Young Joo  Pluriharmonic symbols of commuting Toeplitz type operators on the weighted Bergman spaces A class of Toeplitz type operators acting on the weighted Bergman spaces of the unit ball in the$n$-dimensional complex space is considered and two pluriharmonic symbols of commuting Toeplitz type operators are completely characterized. Keywords:Pluriharmonic functions, Weighted Bergman spaces, Toeplitz type operators.Categories:47B38, 32A37 37. CMB 1998 (vol 41 pp. 10) Borwein, David  Simple conditions for matrices to be bounded operators on$l_p$The two theorems proved yield simple yet reasonably general conditions for triangular matrices to be bounded operators on$l_p$. The theorems are applied to N\"orlund and weighted mean matrices. Keywords:Triangular matrices, NÃ¶rlund matrices, weighted means, operators, on$l_p\$.Categories:47B37, 47A30, 40G05
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