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1. CMB Online first

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 Dunford-Pettis property
Categories:46B20, 46B28, 28B05

2. CMB Online first

Godefroy, G.
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

3. CMB Online first

Swanepoel, Konrad J.
Equilateral sets and a Schütte Theorem for the $4$-norm
A well-known theorem of Schütte (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 Bárány'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

4. 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 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

5. CMB 2012 (vol 56 pp. 503)

Bu, Qingying
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

6. 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 Fréchet 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 Fréchet differentiability, representation
Categories:20M30, 46B10, 46B20, 46E15, 46J10, 49J50

7. 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

8. 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 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)

Lewis, Paul; Schulle, Polly
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)

Kurka, Ondřej
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. 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 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

12. 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

13. CMB 2009 (vol 53 pp. 278)

Galego, Elói M.
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 Pełczyński'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 czyński's decomposition method, Schroeder-Bernstein problem
Categories:46B03, 46B20

14. 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 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)

Ghenciu, Ioana; Lewis, Paul
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 2009 (vol 52 pp. 28)

Choi, Changsun; Kim, Ju Myung; Lee, Keun Young
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

17. CMB 2007 (vol 50 pp. 610)

Rychtář, Jan; Spurný, Jiří
On Weak$^*$ Kadec--Klee Norms
We present partial positive results supporting a conjecture that admitting an equivalent Lipschitz (or uniformly) weak$^*$ Kadec--Klee norm is a three space property.

Keywords:weak$^*$ Kadec--Klee norms, three-space problem
Categories:46B03, 46B2

18. CMB 2007 (vol 50 pp. 519)

Henson, C. Ward; Raynaud, Yves; Rizzo, Andrew
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

19. 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 Ryll-Nardzewski.

Categories:46B20, 46B40, 46B03

20. CMB 2007 (vol 50 pp. 138)

Sari, Bünyamin
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

21. 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 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

22. CMB 2005 (vol 48 pp. 481)

Azagra, D.; Fabian, M.; Jiménez-Sevilla, 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 $C^n$\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

23. CMB 2005 (vol 48 pp. 455)

Rychtář, Jan
On Gâteaux 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:Gâteaux smoothness, Borwein--Preiss variational principle,, weakly compactly generated spaces
Category:46B20

24. CMB 2005 (vol 48 pp. 69)

Fabian, M.; Montesinos, V.; Zizler, V.
Biorthogonal Systems in Weakly Lindelöf 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 sák spaces
Categories:46B03, 46B20., 46B26

25. 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 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
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