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1. CMB Online first
Limited Sets and Bibasic Sequences Bibasic sequences are used to study relative weak compactness
and relative norm compactness of limited sets.
Keywords:limited sets, $L$-sets, bibasic sequences, the Dunford-Pettis property Categories:46B20, 46B28, 28B05 |
2. CMB 2012 (vol 56 pp. 503)
Weak Sequential Completeness of $\mathcal K(X,Y)$ For Banach spaces $X$ and $Y$, we show that if $X^\ast$ and $Y$ are
weakly sequentially complete and every weakly compact operator from
$X$ to $Y$ is compact then the space of all compact operators from $X$
to $Y$ is weakly sequentially complete. The converse is also true if,
in addition, either $X^\ast$ or $Y$ has the bounded compact
approximation property.
Keywords:weak sequential completeness, reflexivity, compact operator space Categories:46B25, 46B28 |
3. CMB 2011 (vol 56 pp. 65)
The Uncomplemented Subspace $\mathbf K(X,Y) $ A vector measure result is used to study the complementation of the
space $K(X,Y)$ of compact operators in the spaces $W(X,Y)$ of weakly
compact operators, $CC(X,Y)$ of completely continuous operators, and
$U(X,Y)$ of unconditionally converging operators.
Results of Kalton and Emmanuele concerning the complementation of
$K(X,Y)$ in $L(X,Y)$ and in $W(X,Y)$ are generalized. The containment
of $c_0$ and $\ell_\infty$ in spaces of operators is also studied.
Keywords:compact operators, weakly compact operators, uncomplemented subspaces of operators Categories:46B20, 46B28 |
4. CMB 2011 (vol 55 pp. 449)
Complemented Subspaces of Linear Bounded Operators We study the complementation of the space $W(X,Y)$ of weakly compact operators, the space $K(X,Y)$ of compact operators, the space $U(X,Y)$ of unconditionally converging operators, and the space $CC(X,Y)$ of completely continuous operators in the space $L(X,Y)$ of bounded linear operators from $X$ to $Y$.
Feder proved that if $X$ is infinite-dimensional and $c_0
\hookrightarrow Y$, then $K(X,Y)$ is uncomplemented in
$L(X,Y)$. Emmanuele and John showed that if $c_0 \hookrightarrow
K(X,Y)$, then $K(X,Y)$ is uncomplemented in $L(X,Y)$.
Bator and Lewis showed that if $X$ is not a Grothendieck space and
$c_0 \hookrightarrow Y$, then $W(X,Y)$ is uncomplemented in
$L(X,Y)$. In this paper, classical results of Kalton and separably
determined operator ideals with property $(*)$ are used to obtain
complementation results that yield these theorems as corollaries.
Keywords:spaces of operators, complemented subspaces, compact operators, weakly compact operators, completely continuous operators Categories:46B20, 46B28 |
5. CMB 2009 (vol 53 pp. 118)
The Uncomplemented Spaces $W(X,Y)$ and $K(X,Y)$ Classical results of Kalton and techniques of Feder are used to study the complementation of the space $W(X, Y)$ of weakly compact operators and the space $K(X,Y)$ of compact operators in the space $L(X,Y)$ of all bounded linear maps from X to Y.
Keywords:spaces of operators, complemented subspace, weakly compact operator, basic sequence Categories:46B28, 46B15, 46B20 |
6. CMB 2009 (vol 52 pp. 213)
Dunford--Pettis Properties and Spaces of Operators J. Elton used an application of Ramsey theory to show that
if $X$ is an infinite dimensional Banach space,
then $c_0$ embeds in $X$, $\ell_1$ embeds in $X$, or there
is a subspace of $X$ that fails to have the Dunford--Pettis property.
Bessaga and Pelczynski showed that if $c_0$ embeds in $X^*$,
then $\ell_\infty$ embeds in $X^*$. Emmanuele and John showed
that if $c_0$ embeds in $K(X,Y)$, then $K(X,Y)$ is not
complemented in $L(X,Y)$. Classical results from Schauder basis theory
are used in a study of Dunford--Pettis sets and strong
Dunford--Pettis sets to extend each of the preceding theorems. The space
$L_{w^*}(X^* , Y)$ of $w^*-w$ continuous operators is also studied.
Keywords:Dunford--Pettis property, Dunford--Pettis set, basic sequence, complemented spaces of operators Categories:46B20, 46B28 |
7. CMB 2009 (vol 52 pp. 28)
Right and Left Weak Approximation Properties in Banach Spaces New necessary and sufficient conditions are established for Banach
spaces to have the approximation property; these conditions are
easier to check than the known ones. A shorter proof of a result
of Grothendieck is presented, and some properties of a weak
version of the approximation property are addressed.
Keywords:approximation property, quasi approximation property, weak approximation property Categories:46B28, 46B10 |