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1. CMB Online first

Yumei, Ma
 Isometry on linear n-G-quasi normed spaces This paper generalizes the Aleksandrov problem: the Mazur-Ulam theorem on $n$-G-quasi normed spaces. It proves that a one-$n$-distance preserving mapping is an $n$-isometry if and only if it has the zero-$n$-G-quasi preserving property, and two kinds of $n$-isometries on $n$-G-quasi normed space are equivalent; we generalize the Benz theorem to n-normed spaces with no restrictions on the dimension of spaces. Keywords:$n$-G-quasi norm, Mazur-Ulam theorem, Aleksandrov problem, $n$-isometry, $n$-0-distanceCategories:46B20, 46B04, 51K05

2. CMB 2016 (vol 59 pp. 769)

García-Pacheco, Francisco Javier; Hill, Justin R.
 Geometric Characterizations of Hilbert Spaces We study some geometric properties related to the set $\Pi_X:= \{ (x,x^* )\in\mathsf{S}_X\times \mathsf{S}_{X^*}:x^* (x )=1 \}$ obtaining two characterizations of Hilbert spaces in the category of Banach spaces. We also compute the distance of a generic element $(h,k )\in H\oplus_2 H$ to $\Pi_H$ for $H$ a Hilbert space. Keywords:Hilbert space, extreme point, smooth, $\mathsf{L}^2$-summandsCategories:46B20, 46C05

3. 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

4. CMB 2014 (vol 58 pp. 150)

Ostrovskii, Mikhail I.
 Connections Between Metric Characterizations of Superreflexivity and the Radon-NikodÃ½ Property for Dual Banach Spaces Johnson and Schechtman (2009) characterized superreflexivity in terms of finite diamond graphs. The present author characterized the Radon-NikodÃ½m property (RNP) for dual spaces in terms of the infinite diamond. This paper is devoted to further study of relations between metric characterizations of superreflexivity and the RNP for dual spaces. The main result is that finite subsets of any set $M$ whose embeddability characterizes the RNP for dual spaces, characterize superreflexivity. It is also observed that the converse statement does not hold, and that $M=\ell_2$ is a counterexample. Keywords:Banach space, diamond graph, finite representability, metric characterization, Radon-NikodÃ½m property, superreflexivityCategories:46B85, 46B07, 46B22

5. 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 spaceCategories:47H09, 46B20, 47H10, 47E10

6. 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 Dunford-Pettis propertyCategories:46B20, 46B28, 28B05

7. 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 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

8. CMB 2013 (vol 57 pp. 640)

 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

9. 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 spacesCategories:46B20, 46C05, 52C17

10. 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 spaceCategories:46B25, 46B28

11. 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, representationCategories:20M30, 46B10, 46B20, 46E15, 46J10, 49J50

12. 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 operatorsCategories:46B20, 46B28

13. 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 operatorsCategories:46B20, 46B28

14. 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 operatorsCategory:46B20

15. 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

16. 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 PeÅczyÅski'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 czyÅski's decomposition method, Schroeder-Bernstein problemCategories:46B03, 46B20

17. 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 sequenceCategories:46B28, 46B15, 46B20

18. 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

19. 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 planeCategories:46B20, 52A21, 52C15

20. 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 operatorsCategories:46B20, 46B28

21. 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 propertyCategories:46B28, 46B10

22. 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

23. 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 problemCategories:46B03, 46B2

24. 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

25. 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
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