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76. 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 estimates
Categories:47B37, 47A30, 40G05

77. 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

78. 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

79. 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, commutant

80. 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.


81. 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

82. 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.


83. 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 operator
Categories:47B38, 30D55, 46E15

84. CMB 1999 (vol 42 pp. 104)

Nikolskaia, Ludmila
Instabilité de vecteurs propres d'opérateurs linéaires
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 kernels
Categories:47A10, 46B15

85. CMB 1999 (vol 42 pp. 87)

Kittaneh, Fuad
Some norm inequalities for operators
Let $A_i$, $B_i$ and $X_i$ $(i=1, 2, \dots, n)$ be operators on a separable Hilbert space. It is shown that if $f$ and $g$ are nonnegative continuous functions on $[0,\infty)$ which satisfy the relation $f(t)g(t) =t$ for all $t$ in $[0,\infty)$, then $$ \Biglvert \,\Bigl|\sum^n_{i=1} A^*_i X_i B_i \Bigr|^r \,\Bigrvert^2 \leq \Biglvert \Bigl( \sum^n_{i=1} A^*_i f (|X^*_i|)^2 A_i \Bigr)^r \Bigrvert \, \Biglvert \Bigl( \sum^n_{i=1} B^*_i g (|X_i|)^2 B_i \Bigr)^r \Bigrvert $$ for every $r>0$ and for every unitarily invariant norm. This result improves some known Cauchy-Schwarz type inequalities. Norm inequalities related to the arithmetic-geometric mean inequality and the classical Heinz inequalities are also obtained.

Keywords:Unitarily invariant norm, positive operator, arithmetic-geometric mean inequality, Cauchy-Schwarz inequality, Heinz inequality
Categories:47A30, 47B10, 47B15, 47B20

86. CMB 1998 (vol 41 pp. 413)

Llorens-Fuster, Enrique; Sims, Brailey
The fixed point property in $\lowercase{c_0}$
A closed convex subset of $c_0$ has the fixed point property ($\fpp$) if every nonexpansive self mapping of it has a fixed point. All nonempty weak compact convex subsets of $c_0$ are known to have the $\fpp$. We show that closed convex subsets with a nonempty interior and nonempty convex subsets which are compact in a topology slightly coarser than the weak topology may fail to have the $\fpp$.

Categories:47H09, 47H10

87. CMB 1998 (vol 41 pp. 434)

Mascioni, Vania; Molnár, Lajos
Linear maps on factors which preserve the extreme points of the unit ball
The aim of this paper is to characterize those linear maps from a von~Neumann factor $\A$ into itself which preserve the extreme points of the unit ball of $\A$. For example, we show that if $\A$ is infinite, then every such linear preserver can be written as a fixed unitary operator times either a unital $\ast$-homomorphism or a unital $\ast$-antihomomorphism.

Categories:47B49, 47D25

88. CMB 1998 (vol 41 pp. 298)

Jahandideh, M. T.
On the ideal-triangularizability of semigroups of quasinilpotent positive operators on $C({\cal K})$
It is known that a semigroup of quasinilpotent integral operators, with positive lower semicontinuous kernels, on $L^2( X, \mu)$, where $X$ is a locally compact Hausdorff-Lindel\"of space and $\mu$ is a $\sigma$-finite regular Borel measure on $X$, is triangularizable. In this article we use the Banach lattice version of triangularizability to establish the ideal-triangularizability of a semigroup of positive quasinilpotent integral operators on $C({\cal K})$ where ${\cal K}$ is a compact Hausdorff space.


89. CMB 1998 (vol 41 pp. 137)

Choksi, J. R.; Nadkarni, M. G.
Genericity of certain classes of unitary and self-adjoint operators
In a paper [1], published in 1990, in a (somewhat inaccessible) conference proceedings, the authors had shown that for the unitary operators on a separable Hilbert space, endowed with the strong operator topology, those with singular, continuous, simple spectrum, with full support, form a dense $G_\delta$. A similar theorem for bounded self-adjoint operators with a given norm bound (omitting simplicity) was recently given by Barry Simon [2], [3], with a totally different proof. In this note we show that a slight modification of our argument, combined with the Cayley transform, gives a proof of Simon's result, with simplicity of the spectrum added.


90. CMB 1998 (vol 41 pp. 240)

Xia, Jingbo
On certain $K$-groups associated with minimal flows
It is known that the Toeplitz algebra associated with any flow which is both minimal and uniquely ergodic always has a trivial $K_1$-group. We show in this note that if the unique ergodicity is dropped, then such $K_1$-group can be non-trivial. Therefore, in the general setting of minimal flows, even the $K$-theoretical index is not sufficient for the classification of Toeplitz operators which are invertible modulo the commutator ideal.

Categories:46L80, 47B35, 47C15

91. CMB 1998 (vol 41 pp. 196)

Nakazi, Takahiko
Brown-Halmos type theorems of weighted Toeplitz operators
The spectra of the Toeplitz operators on the weighted Hardy space $H^2(Wd\th/2\pi)$ and the Hardy space $H^p(d\th/2\pi)$, and the singular integral operators on the Lebesgue space $L^2(d\th/2\pi)$ are studied. For example, the theorems of Brown-Halmos type and Hartman-Wintner type are studied.

Keywords:Toeplitz operator, singular integral, operator, weighted Hardy space, spectrum.

92. 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

93. CMB 1998 (vol 41 pp. 49)

Harrison, K. J.; Ward, J. A.; Eaton, L-J.
Stability of weighted darma filters
We study the stability of linear filters associated with certain types of linear difference equations with variable coefficients. We show that stability is determined by the locations of the poles of a rational transfer function relative to the spectrum of an associated weighted shift operator. The known theory for filters associated with constant-coefficient difference equations is a special case.

Keywords:Difference equations, adaptive $\DARMA$ filters, weighted shifts,, stability and boundedness, automatic continuity
Categories:47A62, 47B37, 93D25, 42A85, 47N70

94. 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, Nörlund matrices, weighted means, operators, on $l_p$.
Categories:47B37, 47A30, 40G05

95. CMB 1997 (vol 40 pp. 443)

Hadwin, Don; Orhon, Mehmet
Reflective Representations and Banach C*-Modules
Suppose ${\cal A}$ is a unital $C$*-algebra and $m\colon{\cal A}\to B(X)$

Categories:47D30, 46L99

96. CMB 1997 (vol 40 pp. 464)

Kuo, Chung-Cheng
On the solvability of a Neumann boundary value problem at resonance
We study the existence of solutions of the semilinear equations (1) $\triangle u + g(x,u)=h$, ${\partial u \over \partial n} = 0$ on $\partial \Omega$ in which the non-linearity $g$ may grow superlinearly in $u$ in one of directions $u \to \infty$ and $u \to -\infty$, and (2) $-\triangle u + g(x,u)=h$, ${\partial u \over \partial n} = 0$ on $\partial \Omega$ in which the nonlinear term $g$ may grow superlinearly in $u$ as $|u| \to \infty$. The purpose of this paper is to obtain solvability theorems for (1) and (2) when the Landesman-Lazer condition does not hold. More precisely, we require that $h$ may satisfy $\int g^\delta_- (x) \, dx < \int h(x) \, dx = 0< \int g^\gamma_+ (x)\,dx$, where $\gamma, \delta$ are arbitrarily nonnegative constants, $g^\gamma_+ (x) = \lim_{u \to \infty} \inf g(x,u) |u|^\gamma$ and $g^\delta_- (x)=\lim_{u \to -\infty} \sup g(x,u)|u|^\delta$. The proofs are based upon degree theoretic arguments.

Keywords:Landesman-Lazer condition, Leray Schauder degree
Categories:35J65, 47H11, 47H15

97. CMB 1997 (vol 40 pp. 193)

Kucerovsky, Dan
Finite rank operators and functional calculus on Hilbert modules over abelian $C^{\ast}$-algebras
We consider the problem: If $K$ is a compact normal operator on a Hilbert module $E$, and $f\in C_0(\Sp K)$ is a function which is zero in a neighbourhood of the origin, is $f(K)$ of finite rank? We show that this is the case if the underlying $C^{\ast}$-algebra is abelian, and that the range of $f(K)$ is contained in a finitely generated projective submodule of $E$.

Categories:55R50, 47A60, 47B38
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