76. CMB 2003 (vol 46 pp. 538)
77. 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 space Categories:46H20, 46H25, 46J10, 46J40, 47L25 

78. CMB 2003 (vol 46 pp. 216)
 Li, ChiKwong; 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 range Categories:47B15, 47B49 

79. 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 operator Categories:32A70, 47G10 

80. CMB 2003 (vol 46 pp. 59)
81. 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 wellknown 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 

82. 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 

83. CMB 2001 (vol 44 pp. 270)
 Cheung, WaiShun; Li, ChiKwong

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 matrices Categories:15A04, 15A60, 47B49 

84. 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 

85. CMB 2000 (vol 43 pp. 157)
86. 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 

87. 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 Category:47A99 

88. CMB 2000 (vol 43 pp. 87)
89. 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
nonzero, 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 

90. CMB 1999 (vol 42 pp. 162)
91. CMB 1999 (vol 42 pp. 139)
92. 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 kernels Categories:47A10, 46B15 

93. 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 CauchySchwarz type inequalities. Norm
inequalities related to the arithmeticgeometric mean inequality and
the classical Heinz inequalities are also obtained.
Keywords:Unitarily invariant norm, positive operator, arithmeticgeometric mean inequality, CauchySchwarz inequality, Heinz inequality Categories:47A30, 47B10, 47B15, 47B20 

94. CMB 1998 (vol 41 pp. 413)
 LlorensFuster, 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 

95. CMB 1998 (vol 41 pp. 434)
96. CMB 1998 (vol 41 pp. 298)
 Jahandideh, M. T.

On the idealtriangularizability 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 HausdorffLindel\"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 idealtriangularizability
of a semigroup of positive quasinilpotent integral operators on
$C({\cal K})$ where ${\cal K}$ is a compact Hausdorff space.
Category:47B65 

97. CMB 1998 (vol 41 pp. 137)
 Choksi, J. R.; Nadkarni, M. G.

Genericity of certain classes of unitary and selfadjoint 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 selfadjoint 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.
Category:47B15 

98. 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 nontrivial. 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 

99. CMB 1998 (vol 41 pp. 196)
 Nakazi, Takahiko

BrownHalmos 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 BrownHalmos type and
HartmanWintner type are studied.
Keywords:Toeplitz operator, singular integral, operator, weighted Hardy space, spectrum. Category:47B35 

100. CMB 1998 (vol 41 pp. 129)