Expand all Collapse all | Results 76 - 95 of 95 |
76. CMB 2000 (vol 43 pp. 193)
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 |
77. CMB 2000 (vol 43 pp. 21)
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 |
78. CMB 2000 (vol 43 pp. 87)
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 |
79. CMB 1999 (vol 42 pp. 452)
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 |
80. CMB 1999 (vol 42 pp. 162)
Lorentz-Schatten Classes and Pointwise Domination of Matrices We investigate pointwise domination property in operator spaces
generated by Lorentz sequence spaces.
Category:47B10 |
81. CMB 1999 (vol 42 pp. 139)
Essential Norm and Weak Compactness of Composition Operators on Weighted Banach Spaces of Analytic Functions |
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 |
82. CMB 1999 (vol 42 pp. 104)
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 |
83. CMB 1999 (vol 42 pp. 87)
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 |
84. CMB 1998 (vol 41 pp. 413)
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 |
85. CMB 1998 (vol 41 pp. 434)
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 |
86. CMB 1998 (vol 41 pp. 298)
On the ideal-triangularizability of semigroups of quasinilpotent positive operators on $C({\cal K})$ |
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.
Category:47B65 |
87. CMB 1998 (vol 41 pp. 137)
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.
Category:47B15 |
88. CMB 1998 (vol 41 pp. 240)
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 |
89. CMB 1998 (vol 41 pp. 196)
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. Category:47B35 |
90. CMB 1998 (vol 41 pp. 129)
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 |
91. CMB 1998 (vol 41 pp. 49)
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 |
92. CMB 1998 (vol 41 pp. 10)
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, NÃ¶rlund matrices, weighted means, operators, on $l_p$. Categories:47B37, 47A30, 40G05 |
93. CMB 1997 (vol 40 pp. 443)
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 |
94. CMB 1997 (vol 40 pp. 464)
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 |
95. CMB 1997 (vol 40 pp. 193)
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 |