This is an announcement for the paper "Entropy numbers of convex hulls
in Banach spaces and applications" by Bernd Carl, Aicke Hinrichs, and
Philipp Rudolph.
Abstract: Entropy numbers and Kolmogorov numbers of convex hulls in
Banach spaces are studied. Applications are given.
Archive classification: math.FA
Mathematics Subject Classification: 41A46, 46B20, 47B06, 46B50
Submitted from: a.hinrichs(a)uni-jena.de
The paper may be downloaded from the archive by web browser from URL
http://front.math.ucdavis.edu/1211.1559
or
http://arXiv.org/abs/1211.1559
This is an announcement for the paper "Quantum expanders and geometry
of operator spaces II" by Gilles Pisier.
Abstract: In this appendix to our paper with the same title posted on
arxiv we give a quick proof of an inequality that can be substituted
to Hastings's result, quoted as Lemma 1.9 in our previous paper. Our
inequality is less sharp but also appears to apply with more general
(and even matricial) coefficients. It shows that up to a universal
constant all moments of the norm of a linear combination of the form
$$S=\sum\nolimits_j a_j U_j \otimes \bar U_j (1-P)$$
are dominated by those of the corresponding Gaussian sum
$$S'=\sum\nolimits_j a_j Y_j \otimes \bar Y'_j .$$
The advantage is that $S'$ is now simply separately a Gaussian random
variable with respect to the independent Gaussian random matrices $(Y_j)$
and $(Y'_j)$, and hence is much easier to majorize. Note we plan to
incorporate this appendix into our future publication.
Archive classification: math.OA
Submitted from: pisier(a)math.jussieu.fr
The paper may be downloaded from the archive by web browser from URL
http://front.math.ucdavis.edu/1211.1055
or
http://arXiv.org/abs/1211.1055
This is an announcement for the paper "Uniform integrability and local
convexity in $L^0$" by Constantinos Kardaras.
Abstract: Let $L^0$ be the vector space of all (equivalence classes of)
real-valued random variables built over a probability space $(\Omega,
\mathcal{F}, P)$, equipped with a metric topology compatible with
convergence in probability. In this work, we provide a necessary
and sufficient structural condition that a set $X \subseteq L^0$
should satisfy in order to infer the existence of a probability
$Q$ that is equivalent to $P$ and such that $X$ is uniformly
$Q$-integrable. Furthermore, we connect the previous essentially
measure-free version of uniform integrability with local convexity of
the $L^0$-topology when restricted on convex, solid and bounded subsets
of $L^0$.
Archive classification: math.FA math.PR
Remarks: 14 pages
Submitted from: langostas(a)gmail.com
The paper may be downloaded from the archive by web browser from URL
http://front.math.ucdavis.edu/1211.0475
or
http://arXiv.org/abs/1211.0475
This is an announcement for the paper "Least squares problems in
orthornormalization" by Shanwen Hu.
Abstract: For any $n$-tuple $(\alpha_1,\cdots,\alpha_n)$ of
linearly independent vectors in Hilbert space $H$, we construct
a unique orthonormal basis $(\epsilon_1,\cdots,\epsilon_n)$
of $span\{\alpha_1,\cdots,\alpha_n\}$ satisfying:
$$\sum_{i=1}^n\|\epsilon_i-\alpha_i\|^2\le\sum_{i=1}^n\|\beta_i-\alpha_i\|^2$$
for all orthonormal basis $(\beta_1,\cdots,\beta_n)$ of
$span\{\alpha_1,\cdots,\alpha_n\}$.
We study the stability of the orthornormalization and give some
applications and examples.
Archive classification: math.FA
Remarks: 10 pages
Submitted from: swhu(a)math.ecnu.edu.cn
The paper may be downloaded from the archive by web browser from URL
http://front.math.ucdavis.edu/1210.7400
or
http://arXiv.org/abs/1210.7400
This is an announcement for the paper "Generalized-Lush Spaces and the
Mazur-Ulam Property" by Dongni Tan, Xujian Huang, and Rui Liu.
Abstract: We introduce a new class of Banach spaces, called
generalized-lush spaces (GL-spaces for short), which contains
almost-CL-spaces, separable lush spaces (specially, separable $C$-rich
subspaces of $C(K)$), and even the two-dimensional space with hexagonal
norm. We obtain that the space $C(K,E)$ of the vector-valued continuous
functions is a GL-space whenever $E$ is, and show that the GL-space
is stable under $c_0$-, $l_1$- and $l_\infty$-sums. As an application,
we prove that the Mazur-Ulam property holds for a larger class of Banach
spaces, called local-GL-spaces, including all lush spaces and GL-spaces.
Furthermore, we generalize the stability properties of GL-spaces to
local-GL-spaces. From this, we can obtain many examples of Banach spaces
having the Mazur-Ulam property.
Archive classification: math.FA
Mathematics Subject Classification: Primary 46B04, Secondary 46B20, 46A22
Remarks: 16 pages
Submitted from: ruiliu(a)nankai.edu.cn
The paper may be downloaded from the archive by web browser from URL
http://front.math.ucdavis.edu/1210.7324
or
http://arXiv.org/abs/1210.7324
This is an announcement for the paper "Slice continuity for operators
and the Daugavet property for bilinear maps" by Enrique A. Sanchez Perez
and Dirk Werner.
Abstract: We introduce and analyse the notion of slice continuity between
operators on Banach spaces in the setting of the Daugavet property.
It is shown that under the slice continuity assumption the Daugavet
equation holds for weakly compact operators. As an application we
define and characterise the Daugavet property for bilinear maps, and
we prove that this allows us to describe some $p$-convexifications of
the Daugavet equation for operators on Banach function spaces that have
recently been introduced.
Archive classification: math.FA
Mathematics Subject Classification: Primary 46B04, secondary 46B25
Submitted from: werner(a)math.fu-berlin.de
The paper may be downloaded from the archive by web browser from URL
http://front.math.ucdavis.edu/1210.7099
or
http://arXiv.org/abs/1210.7099
This is an announcement for the paper "A Bourgain-Pisier construction
for general Banach spaces" by J. Lopez-Abad.
Abstract: We prove that every Banach space, not necessarily separable,
can be isometrically embedded into a $\mathcal L_{\infty}$-space in a
way that the corresponding quotient has the Radon-Nikodym and the Schur
properties. As a consequence, we obtain $\mathcal L_\infty$ spaces
of arbitrary large densities with the Schur and the Radon-Nikodym
properties. This extents the a classical result by J. Bourgain and
G. Pisier.
Archive classification: math.FA
Mathematics Subject Classification: 46B03, 46B26
Submitted from: abad(a)icmat.es
The paper may be downloaded from the archive by web browser from URL
http://front.math.ucdavis.edu/1210.5728
or
http://arXiv.org/abs/1210.5728
This is an announcement for the paper "Applications of fixed point
theorems in the theory of invariant subspaces" by Rafa Espinola and
Miguel Lacruz.
Abstract: We survey several applications of fixed point theorems in the
theory of invariant subspaces. The general idea is that a fixed point
theorem applied to a suitable map yields the existence of invariant
subspaces for an operator on a Banach space.
Archive classification: math.OA
Mathematics Subject Classification: 47A15, 47H10
Remarks: 13 pages, to appear in Fixed Point Theory and Applications
Submitted from: lacruz(a)us.es
The paper may be downloaded from the archive by web browser from URL
http://front.math.ucdavis.edu/1210.5557
or
http://arXiv.org/abs/1210.5557
This is an announcement for the paper "Hadamard differentiability via
Gateaux differentiability" by Ludek Zajicek.
Abstract: Let $X$ be a separable Banach space, $Y$ a Banach space and $f:
X \to Y$ a mapping. We prove that there exists a $\sigma$-directionally
porous set $A\subset X$ such that if $x\in X \setminus A$, $f$ is
Lipschitz at $x$, and $f$ is G\^ateaux differentiable at $x$, then $f$
is Hadamard differentiable at $x$.
If $f$ is Borel measurable (or has the Baire property) and is G\^ ateaux
differentiable at all points, then $f$ is Hadamard differentiable at
all points except a set which is $\sigma$-directionally porous set
(and so is Aronszajn null, Haar null and $\Gamma$-null). Consequently,
an everywhere G\^ ateaux differentiable $f: \R^n \to Y$ is Fr\' echet
differentiable except a nowhere dense $\sigma$-porous set.
Archive classification: math.FA
Mathematics Subject Classification: Primary: 46G05, Secondary: 26B05,
49J50
Remarks: 9 pages
Submitted from: zajicek(a)karlin.mff.cuni.cz
The paper may be downloaded from the archive by web browser from URL
http://front.math.ucdavis.edu/1210.4715
or
http://arXiv.org/abs/1210.4715
This is an announcement for the paper "Mixed integrals and related
inequalities" by Vitali Milman and Liran Rotem.
Abstract: In this paper we define an addition operation on the class
of quasi-concave functions. While the new operation is similar to the
well-known sup-convolution, it has the property that it polarizes the
Lebesgue integral. This allows us to define mixed integrals, which are
the functional analogs of the classic mixed volumes.
We extend various classic inequalities, such as the Brunn-Minkowski and
the Alexandrov-Fenchel inequality, to the functional setting. For
general quasi-concave functions, this is done by restating those results
in the language of rearrangement inequalities. Restricting ourselves
to log-concave functions, we prove generalizations of the Alexandrov
inequalities in a more familiar form.
Archive classification: math.FA math.MG
Mathematics Subject Classification: 52A39, 26B25
Remarks: 30 pages
Submitted from: liranro1(a)post.tau.ac.il
The paper may be downloaded from the archive by web browser from URL
http://front.math.ucdavis.edu/1210.4346
or
http://arXiv.org/abs/1210.4346