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51. CMB 2010 (vol 54 pp. 21)

 Generalized D-symmetric Operators II Let $H$ be a separable, infinite-dimensional, complex Hilbert space and let $A, B\in{\mathcal L }(H)$, where ${\mathcal L}(H)$ is the algebra of all bounded linear operators on $H$. Let $\delta_{AB}\colon {\mathcal L}(H)\rightarrow {\mathcal L}(H)$ denote the generalized derivation $\delta_{AB}(X)=AX-XB$. This note will initiate a study on the class of pairs $(A,B)$ such that $\overline{{\mathcal R}(\delta_{AB})}= \overline{{\mathcal R}(\delta_{A^{\ast}B^{\ast}})}$. Keywords:generalized derivation, adjoint, D-symmetric operator, normal operatorCategories:47B47, 47B10, 47A30

52. CMB 2010 (vol 54 pp. 28)

Chang, Yu-Hsien; Hong, Cheng-Hong
 Generalized Solution of the Photon Transport Problem The purpose of this paper is to show the existence of a generalized solution of the photon transport problem. By means of the theory of equicontinuous $C_{0}$-semigroup on a sequentially complete locally convex topological vector space we show that the perturbed abstract Cauchy problem has a unique solution when the perturbation operator and the forcing term function satisfy certain conditions. A consequence of the abstract result is that it can be directly applied to obtain a generalized solution of the photon transport problem. Keywords:photon transport, $C_{0}$-semigroupCategories:35K30, 47D03

53. CMB 2010 (vol 54 pp. 141)

Kim, Sang Og; Park, Choonkil
 Linear Maps on $C^*$-Algebras Preserving the Set of Operators that are Invertible in $\mathcal{A}/\mathcal{I}$ For $C^*$-algebras $\mathcal{A}$ of real rank zero, we describe linear maps $\phi$ on $\mathcal{A}$ that are surjective up to ideals $\mathcal{I}$, and $\pi(A)$ is invertible in $\mathcal{A}/\mathcal{I}$ if and only if $\pi(\phi(A))$ is invertible in $\mathcal{A}/\mathcal{I}$, where $A\in\mathcal{A}$ and $\pi:\mathcal{A}\to\mathcal{A}/\mathcal{I}$ is the quotient map. We also consider similar linear maps preserving zero products on the Calkin algebra. Keywords:preservers, Jordan automorphisms, invertible operators, zero productsCategories:47B48, 47A10, 46H10

54. CMB 2010 (vol 54 pp. 3)

Bakonyi, M.; Timotin, D.
 Extensions of Positive Definite Functions on Amenable Groups Let $S$ be a subset of an amenable group $G$ such that $e\in S$ and $S^{-1}=S$. The main result of this paper states that if the Cayley graph of $G$ with respect to $S$ has a certain combinatorial property, then every positive definite operator-valued function on $S$ can be extended to a positive definite function on $G$. Several known extension results are obtained as corollaries. New applications are also presented. Categories:43A35, 47A57, 20E05

55. CMB 2010 (vol 53 pp. 550)

Shalit, Orr Moshe
 Representing a Product System Representation as a Contractive Semigroup and Applications to Regular Isometric Dilations In this paper we propose a new technical tool for analyzing representations of Hilbert $C^*$-product systems. Using this tool, we give a new proof that every doubly commuting representation over $\mathbb{N}^k$ has a regular isometric dilation, and we also prove sufficient conditions for the existence of a regular isometric dilation of representations over more general subsemigroups of $\mathbb R_{+}^k$. Categories:47A20, 46L08

56. CMB 2010 (vol 53 pp. 398)

Botelho, Fernanda; Jamison, James
 Projections in the Convex Hull of Surjective Isometries We characterize those linear projections represented as a convex combination of two surjective isometries on standard Banach spaces of continuous functions with values in a strictly convex Banach space. Keywords:isometry, convex combination of isometries, generalized bi-circular projectionsCategories:47A65, 47B15, 47B37

57. CMB 2010 (vol 53 pp. 466)

Dubarbie, Luis
 Separating Maps between Spaces of Vector-Valued Absolutely Continuous Functions In this paper we give a description of separating or disjointness preserving linear bijections on spaces of vector-valued absolutely continuous functions defined on compact subsets of the real line. We obtain that they are continuous and biseparating in the finite-dimensional case. The infinite-dimensional case is also studied. Keywords:separating maps, disjointness preserving, vector-valued absolutely continuous functions, automatic continuityCategories:47B38, 46E15, 46E40, 46H40, 47B33

58. CMB 2008 (vol 51 pp. 604)

{\'S}liwa, Wies{\l}aw
 The Invariant Subspace Problem for Non-Archimedean Banach Spaces It is proved that every infinite-dimensional non-archimedean Banach space of countable type admits a linear continuous operator without a non-trivial closed invariant subspace. This solves a problem stated by A.~C.~M. van Rooij and W.~H. Schikhof in 1992. Keywords:invariant subspaces, non-archimedean Banach spacesCategories:47S10, 46S10, 47A15

59. CMB 2008 (vol 51 pp. 481)

Bayart, Frédéric
 Universal Inner Functions on the Ball It is shown that given any sequence of automorphisms $(\phi_k)_k$ of the unit ball $\bn$ of $\cn$ such that $\|\phi_k(0)\|$ tends to $1$, there exists an inner function $I$ such that the family of non-Euclidean translates" $(I\circ\phi_k)_k$ is locally uniformly dense in the unit ball of $H^\infty(\bn)$. Keywords:inner functions, automorphisms of the ball, universalityCategories:32A35, 30D50, 47B38

60. CMB 2008 (vol 51 pp. 372)

Ezquerro, J. A.; Hernández, M. A.
 Picard's Iterations for Integral Equations of Mixed Hammerstein Type A new semilocal convergence result for the Picard method is presented, where the main required condition in the contraction mapping principle is relaxed. Keywords:nonlinear equations in Banach spaces, successive approximations, semilocal convergence theorem, Picard's iteration, Hammerstein integral equationsCategories:45G10, 47H99, 65J15

61. CMB 2008 (vol 51 pp. 378)

Izuchi, Kou Hei
 Cyclic Vectors in Some Weighted $L^p$ Spaces of Entire Functions In this paper, we generalize a result recently obtained by the author. We characterize the cyclic vectors in $\Lp$. Let $f\in\Lp$ and $f\poly$ be contained in the space. We show that $f$ is non-vanishing if and only if $f$ is cyclic. Keywords:weighted $L^p$ spaces of entire functions, cyclic vectorsCategories:47A16, 46J15, 46H25

62. CMB 2008 (vol 51 pp. 67)

Kalton, Nigel; Sukochev, Fyodor
 Rearrangement-Invariant Functionals with Applications to Traces on Symmetrically Normed Ideals We present a construction of singular rearrangement invariant functionals on Marcinkiewicz function/operator spaces. The functionals constructed differ from all previous examples in the literature in that they fail to be symmetric. In other words, the functional $\phi$ fails the condition that if $x\pprec y$ (Hardy-Littlewood-Polya submajorization) and $0\leq x,y$, then $0\le \phi(x)\le \phi(y).$ We apply our results to singular traces on symmetric operator spaces (in particular on symmetrically-normed ideals of compact operators), answering questions raised by Guido and Isola. Categories:46L52, 47B10, 46E30

63. CMB 2007 (vol 50 pp. 172)

Aron, Richard; Gorkin, Pamela
 An Infinite Dimensional Vector Space of Universal Functions for $H^\infty$ of the Ball We show that there exists a closed infinite dimensional subspace of $H^\infty(B^n)$ such that every function of norm one is universal for some sequence of automorphisms of $B^n$. Categories:47B38, 47B33, 46J10

64. CMB 2007 (vol 50 pp. 85)

Han, Deguang
 Classification of Finite Group-Frames and Super-Frames Given a finite group $G$, we examine the classification of all frame representations of $G$ and the classification of all $G$-frames, \emph{i.e.,} frames induced by group representations of $G$. We show that the exact number of equivalence classes of $G$-frames and the exact number of frame representations can be explicitly calculated. We also discuss how to calculate the largest number $L$ such that there exists an $L$-tuple of strongly disjoint $G$-frames. Keywords:frames, group-frames, frame representations, disjoint framesCategories:42C15, 46C05, 47B10

65. CMB 2006 (vol 49 pp. 117)

Levene, R. H.
 A Double Triangle Operator Algebra From $SL_2(\R)$ We consider the w$^*$-closed operator algebra $\cA_+$ generated by the image of the semigroup $SL_2(\R_+)$ under a unitary representation $\rho$ of $SL_2(\R)$ on the Hilbert~space $L_2(\R)$. We show that $\cA_+$ is a reflexive operator algebra and $\cA_+=\Alg\cD$ where $\cD$ is a double triangle subspace lattice. Surprisingly, $\cA_+$ is also generated as a w$^*$-closed algebra by the image under $\rho$ of a strict subsemigroup of $SL_2(\R_+)$. Categories:46K50, 47L55

66. 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

67. CMB 2005 (vol 48 pp. 409)

Gauthier, P. M.; Xiao, J.
 The Existence of Universal Inner Functions on the Unit Ball of $\mathbb{C}^n$ It is shown that there exists an inner function $I$ defined on the unit ball ${\bf B}^n$ of ${\mathbb C}^n$ such that each function holomorphic on ${\bf B}^n$ and bounded by $1$ can be approximated by non-Euclidean translates" of $I$. Keywords:universal inner functionsCategories:32A35, 30D50, 47B38

68. CMB 2005 (vol 48 pp. 251)

Murphy, G. J.
 The Index Theory Associated to a Non-Finite Trace on a $C^\ast$-Algebra The index theory considered in this paper, a generalisation of the classical Fredholm index theory, is obtained in terms of a non-finite trace on a unital $C^\ast$-algebra. We relate it to the index theory of M.~Breuer, which is developed in a von~Neumann algebra setting, by means of a representation theorem. We show how our new index theory can be used to obtain an index theorem for Toeplitz operators on the compact group $\mathbf{U}(2)$, where the classical index theory does not give any interesting result. Categories:46L, 47B35, 47L80

69. CMB 2005 (vol 48 pp. 97)

Katavolos, Aristides; Paulsen, Vern I.
 On the Ranges of Bimodule Projections We develop a symbol calculus for normal bimodule maps over a masa that is the natural analogue of the Schur product theory. Using this calculus we are easily able to give a complete description of the ranges of contractive normal bimodule idempotents that avoids the theory of J*-algebras. We prove that if $P$ is a normal bimodule idempotent and $\|P\| < 2/\sqrt{3}$ then $P$ is a contraction. We finish with some attempts at extending the symbol calculus to non-normal maps. Categories:46L15, 47L25

70. 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^*$-algebras, summing operators, diagonal operators,, Radon-Nikodym propertyCategories:46L50, 47D15 71. CMB 2004 (vol 47 pp. 504) Cardoso, Fernando; Vodev, Georgi  High Frequency Resolvent Estimates and Energy Decay of Solutions to the Wave Equation We prove an uniform H\"older continuity of the resolvent of the Laplace-Beltrami operator on the real axis for a class of asymptotically Euclidean Riemannian manifolds. As an application we extend a result of Burq on the behaviour of the local energy of solutions to the wave equation. Categories:35B37, 35J15, 47F05 72. CMB 2004 (vol 47 pp. 456) Seto, Michio  On the Berger-Coburn-Lebow Problem for Hardy Submodules In this paper we shall give an affirmative solution to a problem, posed by Berger, Coburn and Lebow, for$C^{\ast}$-algebras on Hardy submodules. Keywords:Hardy submodulesCategory:47B38 73. CMB 2004 (vol 47 pp. 369) Fošner, Ajda; Šemrl, Peter  Spectrally Bounded Linear Maps on${\cal B}(X)$We characterize surjective linear maps on${\cal B}(X)$that are spectrally bounded and spectrally bounded below. Keywords:spectrally bounded linear map.Category:47B49 74. CMB 2004 (vol 47 pp. 257) Marwaha, Alka  A Geometric Characterization of Nonnegative Bands A band is a semigroup of idempotent operators. A nonnegative band$\cls$in$\clb(\cll^2 (\clx))$having at least one element of finite rank and with rank$(S) > 1 $for all$S$in$\cls$is known to have a special kind of common invariant subspace which is termed a standard subspace (defined below). Such bands are called decomposable. Decomposability has helped to understand the structure of nonnegative bands with constant finite rank. In this paper, a geometric characterization of maximal, rank-one, indecomposable nonnegative bands is obtained which facilitates the understanding of their geometric structure. Categories:47D03, 47A15 75. CMB 2004 (vol 47 pp. 298) Yahaghi, Bamdad R.  Near Triangularizability Implies Triangularizability In this paper we consider collections of compact operators on a real or complex Banach space including linear operators on finite-dimensional vector spaces. We show that such a collection is simultaneously triangularizable if and only if it is arbitrarily close to a simultaneously triangularizable collection of compact operators. As an application of these results we obtain an invariant subspace theorem for certain bounded operators. We further prove that in finite dimensions near reducibility implies reducibility whenever the ground field is$\BR$or$\BC\$. Keywords:Linear transformation, Compact operator,, Triangularizability, Banach space, Hilbert, spaceCategories:47A15, 47D03, 20M20
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