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51. 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

52. 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

53. 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

54. CMB 2004 (vol 47 pp. 298)

 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

55. CMB 2004 (vol 47 pp. 215)

Jaworski, Wojciech
 Countable Amenable Identity Excluding Groups A discrete group $G$ is called \emph{identity excluding\/} if the only irreducible unitary representation of $G$ which weakly contains the $1$-dimensional identity representation is the $1$-dimensional identity representation itself. Given a unitary representation $\pi$ of $G$ and a probability measure $\mu$ on $G$, let $P_\mu$ denote the $\mu$-average $\int\pi(g) \mu(dg)$. The goal of this article is twofold: (1)~to study the asymptotic behaviour of the powers $P_\mu^n$, and (2)~to provide a characterization of countable amenable identity excluding groups. We prove that for every adapted probability measure $\mu$ on an identity excluding group and every unitary representation $\pi$ there exists and orthogonal projection $E_\mu$ onto a $\pi$-invariant subspace such that $s$-$\lim_{n\to\infty}\bigl(P_\mu^n- \pi(a)^nE_\mu\bigr)=0$ for every $a\in\supp\mu$. This also remains true for suitably defined identity excluding locally compact groups. We show that the class of countable amenable identity excluding groups coincides with the class of $\FC$-hypercentral groups; in the finitely generated case this is precisely the class of groups of polynomial growth. We also establish that every adapted random walk on a countable amenable identity excluding group is ergodic. Categories:22D10, 22D40, 43A05, 47A35, 60B15, 60J50

56. CMB 2004 (vol 47 pp. 100)

Seto, Michio
 Invariant Subspaces on $\mathbb{T}^N$ and $\mathbb{R}^N$ Let $N$ be an integer which is larger than one. In this paper we study invariant subspaces of $L^2 (\mathbb{T}^N)$ under the double commuting condition. A main result is an $N$-dimensional version of the theorem proved by Mandrekar and Nakazi. As an application of this result, we have an $N$-dimensional version of Lax's theorem. Keywords:invariant subspacesCategories:47A15, 47B47

57. CMB 2004 (vol 47 pp. 144)

Xia, Jingbo
 On the Uniqueness of Wave Operators Associated With Non-Trace Class Perturbations Voiculescu has previously established the uniqueness of the wave operator for the problem of $\mathcal{C}^{(0)}$-perturbation of commuting tuples of self-adjoint operators in the case where the norm ideal $\mathcal{C}$ has the property $\lim_{n\rightarrow\infty} n^{-1/2}\|P_n\|_{\mathcal{C}}=0$, where $\{P_n\}$ is any sequence of orthogonal projections with $\rank(P_n)=n$. We prove that the same uniqueness result holds true so long as $\mathcal{C}$ is not the trace class. (It is well known that there is no such uniqueness in the case of trace-class perturbation.) Categories:47A40, 47B10

58. CMB 2004 (vol 47 pp. 49)

Lindström, Mikael; Makhmutov, Shamil; Taskinen, Jari
 The Essential Norm of a Bloch-to-$Q_p$ Composition Operator The $Q_p$ spaces coincide with the Bloch space for $p>1$ and are subspaces of $\BMOA$ for $0 Keywords:Bloch space, little Bloch space,$\BMOA$,$\VMOA$,$Q_p$spaces,, composition operator, compact operator, essential normCategories:47B38, 47B10, 46E40, 46E15 59. CMB 2003 (vol 46 pp. 538) Borwein, Jonathan; Fitzpatrick, Simon; Girgensohn, Roland  Subdifferentials Whose Graphs Are Not Norm$\times$Weak* Closed In this note we give examples of convex functions whose subdifferentials have unpleasant properties. Particularly, we exhibit a proper lower semicontinuous convex function on a separable Hilbert space such that the graph of its subdifferential is not closed in the product of the norm and bounded weak topologies. We also exhibit a set whose sequential normal cone is not norm closed. Categories:46N10, 47H05 60. CMB 2003 (vol 46 pp. 632) Runde, Volker  The Operator Amenability of Uniform Algebras We prove a quantized version of a theorem by M.~V.~She\u{\i}nberg: A uniform algebra equipped with its canonical, {\it i.e.}, minimal, operator space structure is operator amenable if and only if it is a commutative$C^\ast$-algebra. Keywords:uniform algebras, amenable Banach algebras, operator amenability, minimal, operator spaceCategories:46H20, 46H25, 46J10, 46J40, 47L25 61. CMB 2003 (vol 46 pp. 216) Li, Chi-Kwong; Rodman, Leiba; Šemrl, Peter  Linear Maps on Selfadjoint Operators Preserving Invertibility, Positive Definiteness, Numerical Range Let$H$be a complex Hilbert space, and$\HH$be the real linear space of bounded selfadjoint operators on$H$. We study linear maps$\phi\colon \HH \to \HH$leaving invariant various properties such as invertibility, positive definiteness, numerical range, {\it etc}. The maps$\phi$are not assumed {\it a priori\/} continuous. It is shown that under an appropriate surjective or injective assumption$\phi$has the form$X \mapsto \xi TXT^*$or$X \mapsto \xi TX^tT^*$, for a suitable invertible or unitary$T$and$\xi\in\{1, -1\}$, where$X^t$stands for the transpose of$X$relative to some orthonormal basis. Examples are given to show that the surjective or injective assumption cannot be relaxed. The results are extended to complex linear maps on the algebra of bounded linear operators on$H$. Similar results are proved for the (real) linear space of (selfadjoint) operators of the form$\alpha I+K$, where$\alpha$is a scalar and$K$is compact. Keywords:linear map, selfadjoint operator, invertible, positive definite, numerical rangeCategories:47B15, 47B49 62. CMB 2003 (vol 46 pp. 113) Lee, Jaesung; Rim, Kyung Soo  Properties of the$\mathcal{M}$-Harmonic Conjugate Operator We define the$\mathcal{M}$-harmonic conjugate operator$K$and prove that it is bounded on the nonisotropic Lipschitz space and on$\BMO$. Then we show$K$maps Dini functions into the space of continuous functions on the unit sphere. We also prove the boundedness and compactness properties of$\mathcal{M}$-harmonic conjugate operator with$L^p$symbol. Keywords:$\mathcal{M}$-harmonic conjugate operatorCategories:32A70, 47G10 63. CMB 2003 (vol 46 pp. 59) Constantinescu, T.; Johnson, J. L.  A Note on Noncommutative Interpolation In this paper we formulate and solve Nevanlinna-Pick and Carath\'eodory type problems for tensor algebras with data given on the$N$-dimensional operator unit ball of a Hilbert space. We develop an approach based on the displacement structure theory. Categories:47A57, 47A20 64. CMB 2002 (vol 45 pp. 309) Xia, Jingbo  Joint Mean Oscillation and Local Ideals in the Toeplitz Algebra II: Local Commutivity and Essential Commutant A well-known theorem of Sarason [11] asserts that if$[T_f,T_h]$is compact for every$h \in H^\infty$, then$f \in H^\infty + C(T)$. Using local analysis in the full Toeplitz algebra$\calT = \calT (L^\infty)$, we show that the membership$f \in H^\infty + C(T)$can be inferred from the compactness of a much smaller collection of commutators$[T_f,T_h]$. Using this strengthened result and a theorem of Davidson [2], we construct a proper$C^\ast$-subalgebra$\calT (\calL)$of$\calT$which has the same essential commutant as that of$\calT$. Thus the image of$\calT (\calL)$in the Calkin algebra does not satisfy the double commutant relation [12], [1]. We will also show that no {\it separable} subalgebra$\calS$of$\calT$is capable of conferring the membership$f \in H^\infty + C(T)$through the compactness of the commutators$\{[T_f,S] : S \in \calS\}$. Categories:46H10, 47B35, 47C05 65. CMB 2001 (vol 44 pp. 469) Marcoux, Laurent W.  Sums and Products of Weighted Shifts In this article it is shown that every bounded linear operator on a complex, infinite dimensional, separable Hilbert space is a sum of at most eighteen unilateral (alternatively, bilateral) weighted shifts. As well, we classify products of weighted shifts, as well as sums and limits of the resulting operators. Categories:47B37, 47A99 66. CMB 2001 (vol 44 pp. 270) Cheung, Wai-Shun; Li, Chi-Kwong  Linear Operators Preserving Generalized Numerical Ranges and Radii on Certain Triangular Algebras of Matrices Let$c = (c_1, \dots, c_n)$be such that$c_1 \ge \cdots \ge c_n$. The$c$-numerical range of an$n \times n$matrix$A$is defined by $$W_c(A) = \Bigl\{ \sum_{j=1}^n c_j (Ax_j,x_j) : \{x_1, \dots, x_n\} \text{ an orthonormal basis for } \IC^n \Bigr\},$$ and the$c$-numerical radius of$A$is defined by$r_c (A) = \max \{|z| : z \in W_c (A)\}$. We determine the structure of those linear operators$\phi$on algebras of block triangular matrices, satisfying $$W_c \bigl( \phi(A) \bigr) = W_c (A) \text{ for all } A \quad \text{or} \quad r_c \bigl( \phi(A) \bigr) = r_c (A) \text{ for all } A.$$ Keywords:linear operator, numerical range (radius), block triangular matricesCategories:15A04, 15A60, 47B49 67. 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 68. CMB 2000 (vol 43 pp. 193) Magajna, Bojan  C$^*$-Convexity and the Numerical Range If$A$is a prime C$^*$-algebra,$a \in A$and$\lambda$is in the numerical range$W(a)$of$a$, then for each$\varepsilon > 0$there exists an element$h \in A$such that$\norm{h} = 1$and$\norm{h^* (a-\lambda)h} < \varepsilon$. If$\lambda$is an extreme point of$W(a)$, the same conclusion holds without the assumption that$A$is prime. Given any element$a$in a von Neumann algebra (or in a general C$^*$-algebra)$A$, all normal elements in the weak* closure (the norm closure, respectively) of the C$^*$-convex hull of$a$are characterized. Categories:47A12, 46L05, 46L10 69. CMB 2000 (vol 43 pp. 157) El Abdalaoui, El Houcein  A Larger Class of Ornstein Transformations with Mixing Property We prove that Ornstein transformations are almost surely totally ergodic provided only that the cutting parameter is not bounded. We thus obtain a larger class of Ornstein transformations with the mixing property. Categories:28D05, 47A35 70. CMB 2000 (vol 43 pp. 87) Lindström, Mikael; Schlüchtermann, Georg  Lomonosov's Techniques and Burnside's Theorem In this note we give a proof of Lomonosov's extension of Burnside's theorem to infinite dimensional Banach spaces. Category:47A15 71. CMB 2000 (vol 43 pp. 21) Barnes, Bruce A.  The Commutant of an Abstract Backward Shift A bounded linear operator$T$on a Banach space$X$is an abstract backward shift if the nullspace of$T$is one dimensional, and the union of the null spaces of$T^k$for all$k \geq 1$is dense in$X$. In this paper it is shown that the commutant of an abstract backward shift is an integral domain. This result is used to derive properties of operators in the commutant. Keywords:backward shift, commutantCategory:47A99 72. CMB 1999 (vol 42 pp. 452) Bradley, Sean  Finite Rank Operators in Certain Algebras Let$\Alg(\l)$be the algebra of all bounded linear operators on a normed linear space$\x$leaving invariant each member of the complete lattice of closed subspaces$\l$. We discuss when the subalgebra of finite rank operators in$\Alg(\l)$is non-zero, and give an example which shows this subalgebra may be zero even for finite lattices. We then give a necessary and sufficient lattice condition for decomposing a finite rank operator$F$into a sum of a rank one operator and an operator whose range is smaller than that of$F$, each of which lies in$\Alg(\l)$. This unifies results of Erdos, Longstaff, Lambrou, and Spanoudakis. Finally, we use the existence of finite rank operators in certain algebras to characterize the spectra of Riesz operators (generalizing results of Ringrose and Clauss) and compute the Jacobson radical for closed algebras of Riesz operators and$\Alg(\l)$for various types of lattices. Categories:47D30, 47A15, 47A10 73. CMB 1999 (vol 42 pp. 162) Cobos, Fernando; Kühn, Thomas  Lorentz-Schatten Classes and Pointwise Domination of Matrices We investigate pointwise domination property in operator spaces generated by Lorentz sequence spaces. Category:47B10 74. CMB 1999 (vol 42 pp. 139) Bonet, José; Domański, Paweł; Lindström, Mikael  Essential Norm and Weak Compactness of Composition Operators on Weighted Banach Spaces of Analytic Functions Every weakly compact composition operator between weighted Banach spaces$H_v^{\infty}$of analytic functions with weighted sup-norms is compact. Lower and upper estimates of the essential norm of continuous composition operators are obtained. The norms of the point evaluation functionals on the Banach space$H_v^{\infty}$are also estimated, thus permitting to get new characterizations of compact composition operators between these spaces. Keywords:weighted Banach spaces of holomorphic functions, composition operator, compact operator, weakly compact operatorCategories:47B38, 30D55, 46E15 75. CMB 1999 (vol 42 pp. 104) Nikolskaia, Ludmila  InstabilitÃ© de vecteurs propres d'opÃ©rateurs linÃ©aires We consider some geometric properties of eigenvectors of linear operators on infinite dimensional Hilbert space. It is proved that the property of a family of vectors$(x_n)$to be eigenvectors$Tx_n= \lambda_n x_n$($\lambda_n \noteq \lambda_k$for$n\noteq k$) of a bounded operator$T$(admissibility property) is very instable with respect to additive and linear perturbations. For instance, (1)~for the sequence$(x_n+\epsilon_n v_n)_{n\geq k(\epsilon)}$to be admissible for every admissible$(x_n)$and for a suitable choice of small numbers$\epsilon_n\noteq 0$it is necessary and sufficient that the perturbation sequence be eventually scalar: there exist$\gamma_n\in \C$such that$v_n= \gamma_n v_{k}$for$n\geq k$(Theorem~2); (2)~for a bounded operator$A$to transform admissible families$(x_n)$into admissible families$(Ax_n)$it is necessary and sufficient that$A\$ be left invertible (Theorem~4). Keywords:eigenvectors, minimal families, reproducing kernelsCategories:47A10, 46B15
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