MR2502017 (2010c:46055) Angosto, C.; Cascales, B. Measures of weak noncompactness in Banach spaces. Topology Appl. 156 (2009), no. 7, 1412--1421. (Reviewer: Diana Caponetti) 46B99 (46A50 47B07 47H09 54C35)

The authors consider for a bounded subset H of a Banach space E the De Blasi measure of weak noncompactness w(H) and the measure of weak noncompactness g(H) based on Grothendieck’s double limit criterion. They also deal with the quantitative characteristics k(H) and ck(H) which represent, respectively, the worst distance to E of the weak*-closure of H in the bidual of E and the worst distance to E of the sets of weak*-cluster points in the bidual of E of sequences in H. The authors prove the following chain of inequalities ck(H) < = k(H) < = g(H) < = 2ck(H) < = 2k(H) < = 2w(H), which, in particular, shows that ck, k and g are equivalent. The authors show that ck = k in the class of Banach spaces with Corson property C (i.e, each collection of closed convex subsets of the space with empty intersection has a countable subcollection with empty intersection), but they also give an example for which k(H) = 2ck(H). Moreover, they obtain quantitative counterparts for of Gantmacher’s theorem about weak compactness of adjoint operators in Banach spaces and for the classical Grothendieck’s characterization of weak compactness in spaces C(K)

On k-ball contractive retractions in F-normed ideal spaces

Let X be an infinite dimensional F-normed space and r a positive number such that the closed ball B_r(X) of radius r is properly contained in X. For a bounded subset A of X, the Hausdorff measure of noncompactness gamma(A) of A is the infimum of all \eps >0 such that A has a finite \eps-net in X. A retraction R of B_r(X) onto its boundary is called k-ball contractive if $\gamma(RA) \le k \gamma(A)$ for each subset A of B_r(X). The main aim of this talk is to give examples of regular F-normed ideal spaces in which there is a 1-ball contractive retraction or, for any \eps>0, a (1+ \eps)-ball contractive retraction with positive lower Hausdorff measure of noncompactness

MR2543732 (2010g:46038) Colao, Vittorio; Trombetta, Alessandro; Trombetta, Giulio Hausdorff norms of retractions in Banach spaces of continuous functions. Taiwanese J. Math. 13 (2009), no. 4, 1139–1158. (Reviewer: Diana Caponetti)

A retraction $R$ from the closed unit ball of a Banach space $X$ onto its boundary is called $k$-ball contractive if there is $k \ge 0$ such that $\gamma_X(RA) \le k \gamma_X(A)$ for each subset $A$ of the closed unit ball, where $\gamma_X$ denote the Hausdorff (ball) measure of noncompactness. In the paper under review the authors consider the problem of evaluating the Wo\'{s}ko constant, which is the infimum of all numbers $k$'s for which there is a $k$-ball contractive retraction from the closed unit ball onto the sphere, in Banach spaces of real continuous functions defined on domains which are not necessarily bounded or finite dimensional. The paper extends some previous results valid in spaces of continuous functions to a more general setting. The authors consider the space $\mathcal{B}\mathcal{C}_{B(E)}(K)$ of all real bounded functions which are continuous on $K$ and uniformly continuous on the closed unit ball $B(E)$, being $E$ a normed space and $K$ a set in $E$ containing $B(E)$. They also consider the space ${\mathcal C}(P)$ of all real continuous functions defined on the Hilbert cube $P= \{ x=(x_n) \in l_2 : |x_n| \le \frac1{n} \ \ (n=1,2, ...) \}$. They prove that in both the spaces $\mathcal{B}\mathcal{C}_{B(E)}(K)$ and ${\mathcal C}(P)$ the Wo\'{s}ko constant assumes the smallest possible value $1$, they also give precise estimates of the lower Hausdorff norms and the Hausdorff norms of the retractions they construct

MR2370688 (2009e:46013) Navarro-Pascual, J. C.; Mena-Jurado, J. F.; Sánchez-Lirola, M. G. A two-dimensional inequality and uniformly continuous retractions. J. Math. Anal. Appl. 339 (2008), no. 1, 719--734. (Reviewer: Diana Caponetti) 46B20 (46E40)

Let X be an infinite-dimensional uniformly convex Banach space and let BX and SX be its closed unit ball and unit sphere, respectively. The main result of the paper is that the identity mapping on BX can be expressed as the mean of n uniformly continuous retractions from BX onto SX for every n >= 3. Then, the authors observe that the result holds under a property weaker than uniform convexity, satisfied by any complex Banach space, so that the result generalizes that of [A. Jim´enez-Vargas et al., Studia Math. 135 (1999), no. 1, 75–81; MR1686372 (2000b:46025)]. As an application the extremal structure of spaces of vector-valued uniformly continuous mappings is studied

Recensione: MR2817284 Dhompongsa, S.; Nanan, N. Fixed point theorems by ways of ultra-asymptotic centers. Abstr. Appl. Anal. 2011, Art. ID 826851, 21 pp. (Reviewer: Diana Caponetti

Paper revie

MR2595826 (2011c:46026) Domínguez Benavides, T. The Szlenk index and the fixed point property under renorming. Fixed Point Theory Appl. 2010, Art. ID 268270, 9 pp. (Reviewer: Diana Caponetti)

It is known that not every Banach space can be renormed so that the resultant space satisfies the weak Fixed Point Property (w-FPP). In the paper under review the author gives a further contribution to identify classes of Banach spaces which can be renormed to satisfy the w-FPP. Let $X$ be a Banach space and $X^*$ its dual. The dual norm is $UKK^*$ if for every $\varepsilon >0$ there is $\theta(\varepsilon)>0$ such that every $u$ in the closed unit ball $B_{X^*}$ of $X^*$ with $\|u\| > 1 - \theta(\varepsilon)$ has a weak$^*$ open neighborhood $\mathcal{U}$ with diam$(B_{X^*}\cap\mathcal{U})< \epsilon$. In [Bull. Lond. Math. Soc. 42 (2010), no. 2, 221--228; MR2601548] M. Raya showed that if $X$ is an Asplund space and the Szlenk index $S_z(X) \le \omega$, where $\omega$ denotes the first ordinal number, then there is an equivalent norm on $X$ such that the dual norm on $X^*$ is $UKK^*$. In the paper under review it is proved that whenever $X$ is endowed with this norm, then $R(X) <2$, where R(X)= \sup \{ \lim \inf \|x_n +x\| : x_n \ \mbox{is weakly null with } \ \|x_n\| \le 1, \|x\|=1 \} is the Garc\'ia-Falset“'s coefficient. Since the author and S. Phothi in [Nonlinear Anal. 72 (2010), no. 3-4, 1409-1416; MR2577541] proved that when $X$ is a Banach space which can be continuously embedded in a Banach space $Y$ with $R(Y) <2$, then $X$ can be renormed to satisfy the w-FPP, the results about the Szlenk index lead to the main result of the paper: Let $Y$ be a Banach space with $S_z(Y) \le \omega$, then any Banach space $X$ which can be continuously embedded in $Y$ can be renormed to satisfy the w-FPP. The result applies to Banach spaces which can be continuously embedded in $C(K)$, where $K$ is a scattered compact topological space such that the $\omega$th-derived set $K^{(\omega)}= \emptyset$. In the paper the author also proves that if $(X, \| \cdot\|)$ is a Banach space and $\mathcal{D}$ is the space of all norms in $X$ equivalent to the given one endowed with the metric $\rho(p,q)= \sup \{ |p(x)-q(x)| \}$, where the supremum is taken over all $x$ in the closed unit ball of $X$ and $S_z(X) \le \omega$, then for almost all norms (in the sense of porosity) in $\mathcal{D}$, $X$ satisfies the w-FPP

MR2645846 (2011f:46031) Day, Jerry B.; Lennard, Chris A characterization of the minimal invariant sets of Alspach's mapping. Nonlinear Anal. 73 (2010), no. 1, 221–227. (Reviewer: Diana Caponetti)

Weakly compact, convex subsets in a Banach space need not have the fixed point property for nonexpansive mappings, as shown by D.E. Alspach in [Proc. Amer. Math. Soc. 82 (1981), no. 3, 423–424; MR0612733 (82j:47070)], where the example of a weakly compact, convex subset $C$ of $L_1[0,1]$ and of a nonexpansive self mapping $T$ on $C$ fixed point free is provided. Then, by Zorn's lemma, there exist weakly compact, convex, $T$-invariant fixed point free subsets of the set $C$ which are minimal with respect to these properties. But these minimal invariant sets have not been explicitly characterized. In the paper under review the authors give an explicit formula for the $n$th power $T^n$ of the Alspach's mapping $T$ and they prove that the sequence $(T^nf)$ converges weakly to $\|f\|_1 \chi_{[0,1]}$, for all $f$ in $C$. As a result using [K. Goebel, Concise course on fixed point theorems, Yokohama Publ., Yokohama, 2002; MR1996163 (2004e:47088)] they obtain a description of the minimal invariant sets of the Alspach's mapping $T$. They prove that for all $\alpha \in (0,1)$, Alspach's mapping $T$ is fixed point free on $C_\alpha:= \{f \in C : \|f\|_1 = \alpha \}$, and $\{ D_\infty(\alpha \chi_{[0,1]}) : 0< \alpha <1 \}$ is the collection of all fixed point free minimal invariant subsets of $C$ for $T$, where $D_0(\alpha \chi_{[0,1]}):= \{ \alpha \chi_{[0,1]}\}$, D_{n+1}(\alpha \chi_{[0,1]}):= \mbox{conv} \{ D_n(\alpha \chi_{[0,1]})\cup T( D_n(\alpha \chi_{[0,1]}))\} inductively, and $D_\infty(\alpha \chi_{[0,1]}):= \overline{ \cup_{n=0}^\infty D_n(\alpha \chi_{[0,1]})}$. The authors also give an alternative method to characterize the minimal invariant sets of the Alspach's mapping $T$ which does not require the formula for $T^n$