Expand all Collapse all | Results 1 - 25 of 34 |
1. CMB Online first
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 |
2. 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 |
3. CMB Online first
Uniqueness of preduals in spaces of operators We show that if $E$ is a separable reflexive space, and $L$ is a weak-star closed linear subspace of
$L(E)$ such that $L\cap K(E)$ is weak-star dense in $L$, then $L$ has a unique isometric predual. The proof relies on basic topological arguments.
Categories:46B20, 46B04 |
4. CMB Online first
Equilateral sets and a SchÃ¼tte Theorem for the $4$-norm A well-known 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 |
5. CMB 2012 (vol 57 pp. 42)
Covering the Unit Sphere of Certain Banach Spaces by Sequences of Slices and Balls e prove that, given any covering of any infinite-dimensional 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 point-finite 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 |
6. CMB 2011 (vol 56 pp. 272)
On Super Weakly Compact Convex Sets and Representation of the Dual of the Normed Semigroup They Generate |
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 |
7. 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 |
8. 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 |
9. CMB 2011 (vol 55 pp. 548)
Non-complemented 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 |
10. CMB 2011 (vol 54 pp. 302)
Structure of the Set of Norm-attaining Functionals on Strictly Convex Spaces Let $X$ be a separable non-reflexive Banach space. We show that there
is no Borel class which contains the set of norm-attaining functionals
for every strictly convex renorming of $X$.
Keywords:separable non-reflexive space, set of norm-attaining functionals, strictly convex norm, Borel class Categories:46B20, 54H05, 46B10 |
11. CMB 2009 (vol 53 pp. 278)
Cantor-Bernstein 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 Schroeder--Bernstein (or
Cantor--Bernstein) 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, $\Phi=(p-u)(s-v)-(q+u)(r+v)$ is different from zero and $\Phi$
divides $p+q$ and $r+s$. These sextuples are called Cantor--Bernstein
sextuples for Banach spaces. The simplest case $(1, 0, 0, 1, 1, 1)$
indicates the well-known PeÅczyÅski's decomposition method in
Banach space. On the other hand, by interchanging some Banach spaces
in the above decomposition scheme, refinements of
the Schroeder--Bernstein problem become evident.
Keywords:Pel czyÅski's decomposition method, Schroeder-Bernstein problem Categories:46B03, 46B20 |
12. 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 |
13. CMB 2009 (vol 53 pp. 64)
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 weakly-null 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 |
14. CMB 2009 (vol 52 pp. 424)
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 side-lengths 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 |
15. 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 |
16. CMB 2007 (vol 50 pp. 519)
On Axiomatizability of Non-Commutative $L_p$-Spaces It is shown that Schatten $p$-classes
of operators between Hilbert spaces of different (infinite)
dimensions have ultrapowers which are (completely) isometric to
non-commutative $L_p$-spaces. On the other hand, these Schatten
classes are not themselves isomorphic to non-commutative $L_p$
spaces. As a consequence, the class of non-commutative $L_p$-spaces
is not axiomatizable in the first-order 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 non-commutative $L_p$-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 |
17. CMB 2007 (vol 50 pp. 619)
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
Ryll-Nardzewski.
Categories:46B20, 46B40, 46B03 |
18. CMB 2007 (vol 50 pp. 138)
On the Structure of the Set of Symmetric Sequences in Orlicz Sequence Spaces We study the structure of the sets of symmetric sequences and
spreading models of an Orlicz sequence space equipped with partial
order with respect to domination of bases. In the cases that these
sets are ``small'', some descriptions of the structure of these posets
are obtained.
Categories:46B20, 46B45, 46B07 |
19. CMB 2006 (vol 49 pp. 185)
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 Rogers--Shephard 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 cross-polytope. 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 Banach--Mazur distance to the
regular hexagon.
Keywords:convex body, geometric inequality, thickness, Minkowski space, Banach space, normed space, reduced body, Banach-Mazur compactum, (modified) Banach-Mazur distance, volume ratio Categories:52A40, 46B20 |
20. CMB 2005 (vol 48 pp. 481)
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 $C^n$\nobreakdash-smooth
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
infinite-dimensional, 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 finite-dimensional case, more
restrictive conditions are required. We also study the Fr\'echet
smooth case for mappings from $\mathbb R^n$ to a separable
infinite-dimensional Banach space and the G\^ateaux smooth case
for mappings defined on a separable infinite-dimensional Banach
space and with values in a separable Banach space.
Category:46B20 |
21. CMB 2005 (vol 48 pp. 455)
On GÃ¢teaux Differentiability of Convex Functions in WCG Spaces It is shown, using the Borwein--Preiss 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, Borwein--Preiss variational principle,, weakly compactly generated spaces Category:46B20 |
22. CMB 2005 (vol 48 pp. 69)
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 |
23. CMB 2004 (vol 47 pp. 481)
A New Characterization of Hardy Martingale Cotype Space We give a new characterization of Hardy martingale cotype
property of complex quasi-Banach 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 quasi-Banach space.
Keywords:Hardy martingale, Hardy martingale cotype,, plurisubharmonic function Categories:46B20, 52A07, 60G44 |
24. CMB 2003 (vol 46 pp. 481)
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 |
25. CMB 2003 (vol 46 pp. 242)
Euclidean Sections of Direct Sums of Normed Spaces We study the dimension of ``random'' Euclidean sections of direct sums of
normed spaces. We compare the obtained results with results from \cite{LMS},
to show that for the direct sums the standard randomness with respect to the
Haar measure on Grassmanian coincides with a much ``weaker'' randomness of
``diagonal'' subspaces (Corollary~\ref{sle} and explanation after). We also
add some relative information on ``phase transition''.
Keywords:Dvoretzky theorem, ``random'' Euclidean section, phase transition in asymptotic convexity Categories:46B07, 46B09, 46B20, 52A21 |