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26. CMB 2011 (vol 54 pp. 506)

Neamaty, A.; Mosazadeh, S.
On the Canonical Solution of the Sturm-Liouville Problem with Singularity and Turning Point of Even Order
In this paper, we are going to investigate the canonical property of solutions of systems of differential equations having a singularity and turning point of even order. First, by a replacement, we transform the system to the Sturm-Liouville equation with turning point. Using of the asymptotic estimates provided by Eberhard, Freiling, and Schneider for a special fundamental system of solutions of the Sturm-Liouville equation, we study the infinite product representation of solutions of the systems. Then we transform the Sturm-Liouville equation with turning point to the equation with singularity, then we study the asymptotic behavior of its solutions. Such representations are relevant to the inverse spectral problem.

Keywords:turning point, singularity, Sturm-Liouville, infinite products, Hadamard's theorem, eigenvalues
Categories:34B05, 34Lxx, 47E05

27. CMB 2011 (vol 55 pp. 339)

Loring, Terry A.
From Matrix to Operator Inequalities
We generalize Löwner's method for proving that matrix monotone functions are operator monotone. The relation $x\leq y$ on bounded operators is our model for a definition of $C^{*}$-relations being residually finite dimensional. Our main result is a meta-theorem about theorems involving relations on bounded operators. If we can show there are residually finite dimensional relations involved and verify a technical condition, then such a theorem will follow from its restriction to matrices. Applications are shown regarding norms of exponentials, the norms of commutators, and "positive" noncommutative $*$-polynomials.

Keywords:$C*$-algebras, matrices, bounded operators, relations, operator norm, order, commutator, exponential, residually finite dimensional
Categories:46L05, 47B99

28. CMB 2011 (vol 55 pp. 441)

Zorboska, Nina
Univalently Induced, Closed Range, Composition Operators on the Bloch-type Spaces
While there is a large variety of univalently induced closed range composition operators on the Bloch space, we show that the only univalently induced, closed range, composition operators on the Bloch-type spaces $B^{\alpha}$ with $\alpha \ne 1$ are the ones induced by a disc automorphism.

Keywords:composition operators, Bloch-type spaces, closed range, univalent
Categories:47B35, 32A18

29. CMB 2011 (vol 54 pp. 498)

Mortad, Mohammed Hichem
On the Adjoint and the Closure of the Sum of Two Unbounded Operators
We prove, under some conditions on the domains, that the adjoint of the sum of two unbounded operators is the sum of their adjoints in both Hilbert and Banach space settings. A similar result about the closure of operators is also proved. Some interesting consequences and examples "spice up" the paper.

Keywords:unbounded operators, sum and products of operators, Hilbert and Banach adjoints, self-adjoint operators, closed operators, closure of operators
Category:47A05

30. CMB 2011 (vol 54 pp. 411)

Davidson, Kenneth R.; Wright, Alex
Operator Algebras with Unique Preduals
We show that every free semigroup algebra has a (strongly) unique Banach space predual. We also provide a new simpler proof that a weak-$*$ closed unital operator algebra containing a weak-$*$ dense subalgebra of compact operators has a unique Banach space predual.

Keywords:unique predual, free semigroup algebra, CSL algebra
Categories:47L50, 46B04, 47L35

31. CMB 2011 (vol 54 pp. 255)

Dehaye, Paul-Olivier
On an Identity due to Bump and Diaconis, and Tracy and Widom
A classical question for a Toeplitz matrix with given symbol is how to compute asymptotics for the determinants of its reductions to finite rank. One can also consider how those asymptotics are affected when shifting an initial set of rows and columns (or, equivalently, asymptotics of their minors). Bump and Diaconis obtained a formula for such shifts involving Laguerre polynomials and sums over symmetric groups. They also showed how the Heine identity extends for such minors, which makes this question relevant to Random Matrix Theory. Independently, Tracy and Widom used the Wiener-Hopf factorization to express those shifts in terms of products of infinite matrices. We show directly why those two expressions are equal and uncover some structure in both formulas that was unknown to their authors. We introduce a mysterious differential operator on symmetric functions that is very similar to vertex operators. We show that the Bump-Diaconis-Tracy-Widom identity is a differentiated version of the classical Jacobi-Trudi identity.

Keywords:Toeplitz matrices, Jacobi-Trudi identity, Szegő limit theorem, Heine identity, Wiener-Hopf factorization
Categories:47B35, 05E05, 20G05

32. CMB 2010 (vol 54 pp. 527)

Preda, Ciprian; Sipos, Ciprian
On the Dichotomy of the Evolution Families: A Discrete-Argument Approach
We establish a discrete-time criteria guaranteeing the existence of an exponential dichotomy in the continuous-time behavior of an abstract evolution family. We prove that an evolution family ${\cal U}=\{U(t,s)\}_{t \geq s\geq 0}$ acting on a Banach space $X$ is uniformly exponentially dichotomic (with respect to its continuous-time behavior) if and only if the corresponding difference equation with the inhomogeneous term from a vector-valued Orlicz sequence space $l^\Phi(\mathbb{N}, X)$ admits a solution in the same $l^\Phi(\mathbb{N},X)$. The technique of proof effectively eliminates the continuity hypothesis on the evolution family (\emph{i.e.,} we do not assume that $U(\,\cdot\,,s)x$ or $U(t,\,\cdot\,)x$ is continuous on $[s,\infty)$, and respectively $[0,t]$). Thus, some known results given by Coffman and Schaffer, Perron, and Ta Li are extended.

Keywords:evolution families, exponential dichotomy, Orlicz sequence spaces, admissibility
Categories:34D05, 47D06, 93D20

33. CMB 2010 (vol 54 pp. 364)

Preda, Ciprian; Preda, Petre
Lyapunov Theorems for the Asymptotic Behavior of Evolution Families on the Half-Line
Two theorems regarding the asymptotic behavior of evolution families are established in terms of the solutions of a certain Lyapunov operator equation.

Keywords:evolution families, exponential instability, Lyapunov equation
Categories:34D05, 47D06

34. CMB 2010 (vol 54 pp. 21)

Bouali, S.; Ech-chad, M.
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 operator
Categories:47B47, 47B10, 47A30

35. 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, $C_{0}$-semigroup
Categories:35K30, 47D03

36. 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

37. 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 products
Categories:47B48, 47A10, 46H10

38. 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

39. 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 projections
Categories:47A65, 47B15, 47B37

40. 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 continuity
Categories:47B38, 46E15, 46E40, 46H40, 47B33

41. 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 spaces
Categories:47S10, 46S10, 47A15

42. 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, universality
Categories:32A35, 30D50, 47B38

43. 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 equations
Categories:45G10, 47H99, 65J15

44. 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 vectors
Categories:47A16, 46J15, 46H25

45. 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

46. 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

47. 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 frames
Categories:42C15, 46C05, 47B10

48. 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

49. 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 theory
Categories:47B35, 46L80

50. 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 functions
Categories:32A35, 30D50, 47B38
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