1. CMB 2015 (vol 59 pp. 3)
 Alfuraidan, Monther Rashed

The Contraction Principle for Multivalued Mappings on a Modular Metric Space with a Graph
We study the existence of fixed points for contraction multivalued
mappings in modular metric spaces endowed with a graph. The
notion of a modular metric on an arbitrary set and the corresponding
modular spaces, generalizing classical modulars over linear spaces
like Orlicz spaces, were recently introduced. This paper can
be seen as a generalization of Nadler's and Edelstein's fixed
point theorems to modular metric spaces endowed with a graph.
Keywords:fixed point theory, modular metric spaces, multivalued contraction mapping, connected digraph. Categories:47H09, 46B20, 47H10, 47E10 

2. CMB 2014 (vol 58 pp. 297)
 Khamsi, M. A.

Approximate Fixed Point Sequences of Nonlinear Semigroup in Metric Spaces
In this paper, we investigate the common
approximate fixed point sequences of nonexpansive semigroups of
nonlinear mappings $\{T_t\}_{t \geq 0}$, i.e., a family such that
$T_0(x)=x$, $T_{s+t}=T_s(T_t(x))$, where the domain is a metric space
$(M,d)$. In particular we prove that under suitable conditions, the
common approximate fixed point sequences set is the same as the common
approximate fixed point sequences set of two mappings from the family.
Then we use the Ishikawa iteration to construct a common approximate
fixed point sequence of nonexpansive semigroups of nonlinear
mappings.
Keywords:approximate fixed point, fixed point, hyperbolic metric space, Ishikawa iterations, nonexpansive mapping, semigroup of mappings, uniformly convex hyperbolic space Categories:47H09, 46B20, 47H10, 47E10 

3. CMB 2014 (vol 58 pp. 71)
 Ghenciu, Ioana

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 DunfordPettis property Categories:46B20, 46B28, 28B05 

4. CMB 2014 (vol 57 pp. 810)
 Godefroy, G.

Uniqueness of Preduals in Spaces of Operators
We show that if $E$ is a separable reflexive space, and $L$ is a weakstar closed linear subspace of
$L(E)$ such that $L\cap K(E)$ is weakstar dense in $L$, then $L$ has a unique isometric predual. The proof relies on basic topological arguments.
Categories:46B20, 46B04 

5. CMB 2013 (vol 57 pp. 640)
 Swanepoel, Konrad J.

Equilateral Sets and a SchÃ¼tte Theorem for the $4$norm
A wellknown theorem of SchÃ¼tte (1963) gives a sharp lower bound for
the ratio of the maximum and minimum distances between $n+2$ points in
$n$dimensional Euclidean space.
In this note we adapt BÃ¡rÃ¡ny's elegant proof (1994) of this theorem to the space $\ell_4^n$.
This gives a new proof that the largest cardinality of an equilateral
set in $\ell_4^n$ is $n+1$, and gives a constructive bound for an
interval $(4\varepsilon_n,4+\varepsilon_n)$ of values of $p$ close to $4$ for which it is known that the largest cardinality of an equilateral set in $\ell_p^n$ is $n+1$.
Categories:46B20, 52A21, 52C17 

6. CMB 2012 (vol 57 pp. 42)
 Fonf, Vladimir P.; Zanco, Clemente

Covering the Unit Sphere of Certain Banach Spaces by Sequences of Slices and Balls
e prove that, given any covering of any infinitedimensional Hilbert space $H$ by countably many closed balls, some point exists in $H$ which belongs to infinitely many balls. We do that by characterizing isomorphically polyhedral separable Banach spaces as those whose unit sphere admits a pointfinite covering by the union of countably many slices of the unit ball.
Keywords:point finite coverings, slices, polyhedral spaces, Hilbert spaces Categories:46B20, 46C05, 52C17 

7. CMB 2011 (vol 56 pp. 272)
 Cheng, Lixin; Luo, Zhenghua; Zhou, Yu

On Super Weakly Compact Convex Sets and Representation of the Dual of the Normed Semigroup They Generate
In this note, we first give a characterization of super weakly
compact convex sets of a Banach space $X$:
a closed bounded convex set $K\subset X$ is
super weakly compact if and only if there exists a $w^*$ lower
semicontinuous seminorm $p$ with $p\geq\sigma_K\equiv\sup_{x\in
K}\langle\,\cdot\,,x\rangle$ such that $p^2$ is uniformly FrÃ©chet
differentiable on each bounded set of $X^*$. Then we present a
representation theorem for the dual of the semigroup $\textrm{swcc}(X)$
consisting of all the nonempty super weakly compact convex sets of the
space $X$.
Keywords:super weakly compact set, dual of normed semigroup, uniform FrÃ©chet differentiability, representation Categories:20M30, 46B10, 46B20, 46E15, 46J10, 49J50 

8. CMB 2011 (vol 56 pp. 65)
 Ghenciu, Ioana

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 

9. CMB 2011 (vol 55 pp. 449)
 Bahreini, Manijeh; Bator, Elizabeth; Ghenciu, Ioana

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 infinitedimensional 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 

10. CMB 2011 (vol 55 pp. 548)
 Lewis, Paul; Schulle, Polly

Noncomplemented Spaces of Operators, Vector Measures, and $c_o$
The Banach spaces $L(X, Y)$, $K(X, Y)$, $L_{w^*}(X^*, Y)$, and
$K_{w^*}(X^*, Y)$ are studied to determine when they contain the
classical Banach spaces $c_o$ or $\ell_\infty$. The complementation of
the Banach space $K(X, Y)$ in $L(X, Y)$ is discussed as well as what
impact this complementation has on the embedding of $c_o$ or
$\ell_\infty$ in $K(X, Y)$ or $L(X, Y)$. Results of Kalton, Feder, and
Emmanuele concerning the complementation of $K(X, Y)$ in $L(X, Y)$ are
generalized. Results concerning the complementation of the Banach
space $K_{w^*}(X^*, Y)$ in $L_{w^*}(X^*, Y)$ are also explored as well
as how that complementation affects the embedding of $c_o$ or
$\ell_\infty$ in $K_{w^*}(X^*, Y)$ or $L_{w^*}(X^*, Y)$. The $\ell_p$
spaces for $1 = p < \infty$ are studied to determine when the space of
compact operators from one $\ell_p$ space to another contains
$c_o$. The paper contains a new result which classifies these spaces
of operators. A new result using vector measures is given to
provide more efficient proofs of theorems by Kalton, Feder, Emmanuele,
Emmanuele and John, and Bator and Lewis.
Keywords:spaces of operators, compact operators, complemented subspaces, $w^*w$compact operators Category:46B20 

11. CMB 2011 (vol 54 pp. 302)
12. CMB 2009 (vol 53 pp. 278)
 Galego, Elói M.

CantorBernstein Sextuples for Banach Spaces
Let $X$ and $Y$ be Banach spaces isomorphic
to complemented subspaces of each other with supplements $A$ and
$B$. In 1996, W. T. Gowers solved the SchroederBernstein (or
CantorBernstein) problem for Banach spaces by showing that $X$ is not
necessarily isomorphic to $Y$. In this paper, we obtain a necessary
and sufficient condition on the sextuples $(p, q, r, s, u, v)$ in
$\mathbb N$
with $p+q \geq 1$, $r+s \geq 1$ and $u, v \in \mathbb N^*$, to provide that
$X$ is isomorphic to $Y$, whenever these spaces satisfy the following
decomposition scheme
$$
A^u \sim X^p \oplus Y^q, \quad
B^v \sim X^r \oplus Y^s.
$$
Namely, $\Phi=(pu)(sv)(q+u)(r+v)$ is different from zero and $\Phi$
divides $p+q$ and $r+s$. These sextuples are called CantorBernstein
sextuples for Banach spaces. The simplest case $(1, 0, 0, 1, 1, 1)$
indicates the wellknown PeÅczyÅski's decomposition method in
Banach space. On the other hand, by interchanging some Banach spaces
in the above decomposition scheme, refinements of
the SchroederBernstein problem become evident.
Keywords:Pel czyÅski's decomposition method, SchroederBernstein problem Categories:46B03, 46B20 

13. CMB 2009 (vol 53 pp. 118)
 Lewis, Paul

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 

14. CMB 2009 (vol 53 pp. 64)
 Dodos, Pandelis

On Antichains of Spreading Models of Banach Spaces
We show that for every separable Banach space $X$,
either $\mathrm{SP_w}(X)$ (the set of all spreading models
of $X$ generated by weaklynull sequences in $X$, modulo
equivalence) is countable, or $\mathrm{SP_w}(X)$ contains an
antichain of the size of the continuum. This answers
a question of S.~J. Dilworth, E. Odell, and B. Sari.
Categories:46B20, 03E15 

15. CMB 2009 (vol 52 pp. 424)
 Martini, Horst; Spirova, Margarita

Covering Discs in Minkowski Planes
We investigate the following version of the circle covering
problem in strictly convex (normed or) Minkowski planes: to cover
a circle of largest possible diameter by $k$ unit circles. In
particular, we study the cases $k=3$, $k=4$, and $k=7$. For $k=3$
and $k=4$, the diameters under consideration are described in
terms of sidelengths and circumradii of certain inscribed regular
triangles or quadrangles. This yields also simple explanations of
geometric meanings that the corresponding homothety ratios have.
It turns out that basic notions from Minkowski geometry play an
essential role in our proofs, namely Minkowskian bisectors,
$d$segments, and the monotonicity lemma.
Keywords:affine regular polygon, bisector, circle covering problem, circumradius, $d$segment, Minkowski plane, (strictly convex) normed plane Categories:46B20, 52A21, 52C15 

16. CMB 2009 (vol 52 pp. 213)
 Ghenciu, Ioana; Lewis, Paul

DunfordPettis 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 DunfordPettis 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 DunfordPettis sets and strong
DunfordPettis sets to extend each of the preceding theorems. The space
$L_{w^*}(X^* , Y)$ of $w^*w$ continuous operators is also studied.
Keywords:DunfordPettis property, DunfordPettis set, basic sequence, complemented spaces of operators Categories:46B20, 46B28 

17. CMB 2007 (vol 50 pp. 519)
 Henson, C. Ward; Raynaud, Yves; Rizzo, Andrew

On Axiomatizability of NonCommutative $L_p$Spaces
It is shown that Schatten $p$classes
of operators between Hilbert spaces of different (infinite)
dimensions have ultrapowers which are (completely) isometric to
noncommutative $L_p$spaces. On the other hand, these Schatten
classes are not themselves isomorphic to noncommutative $L_p$
spaces. As a consequence, the class of noncommutative $L_p$spaces
is not axiomatizable in the firstorder language developed by
Henson and Iovino for normed space structures, neither in the
signature of Banach spaces, nor in that of operator spaces. Other
examples of the same phenomenon are presented that belong to the
class of corners of noncommutative $L_p$spaces. For $p=1$ this
last class, which is the same as the class of preduals of ternary
rings of operators, is itself axiomatizable in the signature of
operator spaces.
Categories:46L52, 03C65, 46B20, 46L07, 46M07 

18. CMB 2007 (vol 50 pp. 619)
 Tcaciuc, Adi

On the Existence of Asymptotic$l_p$ Structures in Banach Spaces
It is shown that if a Banach space is saturated with infinite
dimensional subspaces in which all ``special" $n$tuples of
vectors are equivalent with constants independent of $n$tuples and
of $n$, then the space contains asymptotic$l_p$ subspaces
for some $1 \leq p \leq \infty$.
This extends a result by Figiel, Frankiewicz, Komorowski and
RyllNardzewski.
Categories:46B20, 46B40, 46B03 

19. CMB 2007 (vol 50 pp. 138)
20. CMB 2006 (vol 49 pp. 185)
 Averkov, Gennadiy

On the Inequality for Volume and Minkowskian Thickness
Given a centrally symmetric convex body $B$ in $\E^d,$ we denote
by $\M^d(B)$ the Minkowski space ({\em i.e.,} finite dimensional
Banach space) with unit ball $B.$ Let $K$ be an arbitrary convex
body in $\M^d(B).$ The relationship between volume $V(K)$ and the
Minkowskian thickness ($=$ minimal width) $\thns_B(K)$ of $K$ can
naturally be given by the sharp geometric inequality $V(K) \ge
\alpha(B) \cdot \thns_B(K)^d,$ where $\alpha(B)>0.$ As a simple
corollary of the RogersShephard inequality we obtain that
$\binom{2d}{d}{}^{1} \le \alpha(B)/V(B) \le 2^{d}$ with equality
on the left attained if and only if $B$ is the difference body of
a simplex and on the right if $B$ is a crosspolytope. The main
result of this paper is that for $d=2$ the equality on the right
implies that $B$ is a parallelogram. The obtained results yield
the sharp upper bound for the modified BanachMazur distance to the
regular hexagon.
Keywords:convex body, geometric inequality, thickness, Minkowski space, Banach space, normed space, reduced body, BanachMazur compactum, (modified) BanachMazur distance, volume ratio Categories:52A40, 46B20 

21. CMB 2005 (vol 48 pp. 481)
 Azagra, D.; Fabian, M.; JiménezSevilla, M.

Exact Filling of Figures with the Derivatives of Smooth Mappings Between Banach Spaces
We establish sufficient conditions on the shape of a set $A$
included in the space $\mathcal L _s^n(X,Y)$ of the $n$linear
symmetric mappings between Banach spaces $X$ and $Y$, to ensure
the existence of a $C^n$\nobreakdashsmooth
mapping $f\colon X \rightarrow Y$,
with bounded support, and such that $f^{(n)}(X)=A$, provided that
$X$ admits a $C^{n}$smooth bump with bounded $n$th derivative
and $\dens X=\dens \mathcal L ^n(X,Y)$. For instance, when $X$ is
infinitedimensional, every bounded connected and open set $U$
containing the origin is the range of the $n$th derivative of
such a mapping. The same holds true for the closure of $U$,
provided that every point in the boundary of $U$ is the end
point of a path within $U$. In the finitedimensional case, more
restrictive conditions are required. We also study the Fr\'echet
smooth case for mappings from $\mathbb R^n$ to a separable
infinitedimensional Banach space and the G\^ateaux smooth case
for mappings defined on a separable infinitedimensional Banach
space and with values in a separable Banach space.
Category:46B20 

22. CMB 2005 (vol 48 pp. 455)
 Rychtář, Jan

On GÃ¢teaux Differentiability of Convex Functions in WCG Spaces
It is shown, using the BorweinPreiss variational principle
that for every continuous convex function $f$ on
a weakly compactly generated space $X$,
every $x_0\in X$ and every weakly compact convex symmetric set $K$ such that
$\cspan K=X$,
there is a point of G\^ateaux differentiability of $f$ in $x_0+K$.
This extends a Klee's result for separable spaces.
Keywords:GÃ¢teaux smoothness, BorweinPreiss variational principle,, weakly compactly generated spaces Category:46B20 

23. CMB 2005 (vol 48 pp. 69)
 Fabian, M.; Montesinos, V.; Zizler, V.

Biorthogonal Systems in Weakly LindelÃ¶f Spaces
We study countable splitting of Markushevich bases in weakly Lindel\"of
Banach spaces in connection with the geometry of these spaces.
Keywords:Weak compactness, projectional resolutions,, Markushevich bases, Eberlein compacts, Va\v sÃ¡k spaces Categories:46B03, 46B20., 46B26 

24. CMB 2004 (vol 47 pp. 481)
 Bekjan, Turdebek N.

A New Characterization of Hardy Martingale Cotype Space
We give a new characterization of Hardy martingale cotype
property of complex quasiBanach space by using the existence of a
kind of plurisubharmonic functions. We also characterize the best
constants of Hardy martingale inequalities with values
in the complex quasiBanach space.
Keywords:Hardy martingale, Hardy martingale cotype,, plurisubharmonic function Categories:46B20, 52A07, 60G44 

25. CMB 2003 (vol 46 pp. 481)
 Bachir, M.; Lancien, G.

On the Composition of Differentiable Functions
We prove that a Banach space $X$ has the Schur property if and only if every
$X$valued weakly differentiable function is Fr\'echet differentiable. We
give a general result on the Fr\'echet differentiability of $f\circ T$, where
$f$ is a Lipschitz function and $T$ is a compact linear operator. Finally
we study, using in particular a smooth variational principle, the
differentiability of the semi norm $\Vert \ \Vert_{\lip}$ on various spaces
of Lipschitz functions.
Categories:58C20, 46B20 
