Geometry Of Banach Spaces Research Articles

ON ASYMPTOTIC UNIFORM SMOOTHNESS AND NONLINEAR GEOMETRY OF BANACH SPACES

AbstractThese notes concern the nonlinear geometry of Banach spaces, asymptotic uniform smoothness and several Banach–Saks-like properties. We study the existence of certain concentration inequalities in asymptotically uniformly smooth Banach spaces as well as weakly sequentially continuous coarse (Lipschitz) embeddings into those spaces. Some results concerning the descriptive set theoretical complexity of those properties are also obtained. We finish the paper with a list of open problem.

The Geometry of Projective, Injective, and Flat Banach Modules

In this paper, we prove general facts on metrically and topologically projective, injective, and flat Banach modules. We prove theorems pointing to the close connection between metric, topological Banach homology and the geometry of Banach spaces. For example, in geometric terms we give a complete description of projective, injective, and flat annihilator modules. We also show that for an algebra with the geometric structure of an - or -space all its homologically trivial modules possess the Dunford–Pettis property.

To history of influence of the theorem of Milyutin on researches in geometry of Banach spaces

To history of influence of the theorem of Milyutin on researches in geometry of Banach spaces

Coupling of Brownian motions in Banach spaces

Consider a separable Banach space $\mathcal{W} $ supporting a non-trivial Gaussian measure $\mu $. The following is an immediate consequence of the theory of Gaussian measure on Banach spaces: there exist (almost surely) successful couplings of two $\mathcal{W} $-valued Brownian motions $\mathbf{B} $ and $\widetilde{\mathbf {B}} $ begun at starting points $\mathbf{B} (0)$ and $\widetilde{\mathbf {B}} (0)$ if and only if the difference $\mathbf{B} (0)-\widetilde{\mathbf {B}} (0)$ of their initial positions belongs to the Cameron-Martin space $\mathcal{H} _\mu $ of $\mathcal{W} $ corresponding to $\mu $. For more general starting points, can there be a “coupling at time $\infty $”, such that almost surely $\|{\mathbf {B}(t)-\widetilde {\mathbf {B}}(t)}\|_{\mathcal{W} } \to 0$ as $t\to \infty $? Such couplings exist if there exists a Schauder basis of $\mathcal{W} $ which is also a $\mathcal{H} _\mu $-orthonormal basis of $\mathcal{H} _\mu $. We propose (and discuss some partial answers to) the question, to what extent can one express the probabilistic Banach space property “Brownian coupling at time $\infty $ is always possible” purely in terms of Banach space geometry?

Abstract Characterization of a Conditional Expectation Operator on the Space of Measurable Sections

A conditional expectation operator plays an important role in geometry of Banach spaces. However, the main issue is with regards to the existence of a conditional expectation operator that permits other objects to be considered such as martingales and martingale convergence theorems. Thus, the purpose of this study is to provide an abstract characterization of a conditional expectation operator on a space of measurable sections.

Asymptotic structure and coarse Lipschitz geometry of Banach spaces

In this paper, we study the coarse Lipschitz geometry of Banach spaces with several asymptotic properties. Specifically, we look at asymptotically uniformly smoothness and convexity, and several distinct Banach-Saks-like properties. Among other results, we characterize the Banach spaces which are either coarsely or uniformly homeomorphic to $T^{p_1}\oplus \ldots \oplus T^{p_n}$, where each $T^{p_j}$ denotes the $p_j$-convexification of the Tsirelson space, for $p_1,\ldots,p_n\in (1,\ldots, \infty)$, and $2\not\in\{p_1,\ldots ,p_n\}$. We obtain applications to the coarse Lipschitz geometry of the $p$-convexifications of the Schlumprecht space, and some hereditarily indecomposable Banach spaces. We also obtain some new results on the linear theory of Banach spaces.

BIRKHOFF ORTHOGONALITY IN CLASSICAL -IDEALS

The Birkhoff orthogonality has been recently intensively studied in connection with the geometry of Banach spaces and operator theory. The main aim of this paper is to characterize the Birkhoff orthogonality in ${\mathcal{L}}(X;Y)$ under the assumption that ${\mathcal{K}}(X;Y)$ is an $M$-ideal in ${\mathcal{L}}(X;Y)$. Moreover, we survey the known results, as well as giving some new and more general ones. Furthermore, we characterize an approximate Birkhoff orthogonality in ${\mathcal{K}}(X;Y)$.

Near dentability and continuity of the set-valued metric generalized inverse in Banach spaces

In this paper, upper semicontinuity and continuity of the set-valued metric generalized inverse T∂ in nearly dentable spaces are investigated using the methods of Banach space geometry. Moreover, it is proved that if X is a nearly dentable space and if C is a closed convex set of X, then C is approximatively compact if and only if PC(x) is compact for any x∈X.

Henryk Hudzik – vita et opera

This article contains a short vita of Henryk Hudzik's as well as a non-exhaustive survey of his contribution to various areas of analysis. We focus on the theory of Orlicz−Sobolev spaces and the geometry of Banach spaces. We highlight criteria for some important geometric properties related to the metric fixed point theory in some classes of Banach lattices, including Orlicz and Orlicz−Lorentz spaces, but we do not forget Henryk Hudzik's contribution to nonlinear integral equations and partial differential equations.

On Some Applications of Geometry of Banach Spaces and Some New Results Related to the Fixed Point Theory in Orlicz Sequence Spaces

On Some Applications of Geometry of Banach Spaces and Some New Results Related to the Fixed Point Theory in Orlicz Sequence Spaces

Applied Harmonic Analysis and Sparse Approximation

Efficiently analyzing functions, in particular multivariate functions, is a key problem in applied mathematics. The area of applied harmonic analysis has a significant impact on this problem by providing methodologies both for theoretical questions and for a wide range of applications in technology and science, such as image processing. Approximation theory, in particular the branch of the theory of sparse approximations, is closely intertwined with this area with a lot of recent exciting developments in the intersection of both. Research topics typically also involve related areas such as convex optimization, probability theory, and Banach space geometry. The workshop was the continuation of a first event in 2012 and intended to bring together world leading experts in these areas, to report on recent developments, and to foster new developments and collaborations.

Quantitative affine approximation for UMD targets

Quantitative affine approximation for UMD targets, Discrete Analysis 2016:6, 48 pp. Let $Y$ be a Banach space. A _martingale difference sequence_ in $Y$ is a sequence of $Y$-valued random variables such that $\mathbb{E}[d_i|d_1,\dots,d_{i-1}]=0$ for every $i$ (and $\mathbb{E}(d_1)=0$). Given $1<p<\infty$, we say that $Y$ satisfies the UMD$_p$ property if there is a constant $C_p$ such that $$\mathbb{E}\|\sum_{i=1}^n\epsilon_id_i\|^p\leq C_p^p\mathbb{E}\|\sum_{i=1}^nd_i\|^p$$ for every martingale difference sequence $d_1,\dots,d_n$ and every choice of signs $\epsilon_1,\dots,\epsilon_n\in\{-1,1_{}\}$. An _unconditional martingale difference_ space, or UMD space for short, is a Banach space $Y$ that satisfies the UMD$_p$ property for some $1<p<\infty$, which can be shown to imply that it satisfies the UMD$_p$ property for all such $p$. It is easy to show that when $Y$ is a Hilbert space, it has the UMD$_p$ property with a constant of 1. In the context of stochastic analysis, UMD spaces can be thought of as spaces that have many of the good properties of Hilbert spaces. The question considered in this paper concerns a kind of differentiability property of Lipschitz functions. Let $X$ be an $n$-dimensional normed space with unit ball $B_X$ and let $Y$ be a UMD Banach space. Given a 1-Lipschitz function $f:B_X\to Y$ and $\epsilon>0$, the aim is to find a sub-ball $B'\subset B$ of radius $\rho$, with $\rho$ as large as possible, and an affine function $\Lambda:B'\to Y$, such that $\|f(x)-\Lambda(x)\|\leq\epsilon\rho$ for every $x\in B'$. Importantly, the lower bound on $\rho$ must be independent of the function $f$. The paper obtains a lower bound of $\exp(-(1/\epsilon)^{cn})$ for a constant $c$ that depends on $Y$ only. This improves on the previously best known bound even when $Y$ is a Hilbert space. In the other direction, the best known upper bound (on how large $\rho$ is in the worst case) is singly exponential in $n$, so there is still a large and very interesting gap. There are several other attractive open problems in the paper. The proof of the main result of the paper is not easy, but it contains many new ideas in the geometry of Banach spaces and in harmonic analysis, some of which are interesting results in their own right.

Equivalent norms with the property $$(\beta )$$ of Rolewicz

We extend to the non separable setting many characterizations of the Banach spaces admitting an equivalent norm with the property \((\beta )\) of Rolewicz. These characterizations involve in particular the Szlenk index and asymptotically uniformly smooth or convex norms. This allows to extend easily to the non separable case some recent results from the non linear geometry of Banach spaces.

Characterization of approximative compactness in Banach spaces

In this paper, we rst prove that approximative compactness and near dentability are equivalent for a Banach space. Then, we present several characterizations that every weak* closed convex subset with non-empty weak* interior in a dual space is either approximatively compact or weakly approximatively compact. Our proofs are based on some subtle techniques in geometry of Banach spaces.

Volume Ratio, Sparsity, and Minimaxity Under Unitarily Invariant Norms

This paper studies non-asymptotic minimax estimation of high-dimensional matrices and provides tight minimax rates for a large collection of loss functions in a variety of problems via information-theoretic methods. Based on the convex geometry of finite-dimensional Banach spaces, we first develop a volume ratio approach for determining minimax estimation rates of unconstrained mean matrices under all unitarily invariant norm losses, which turn out to only depend on the norm of identity matrix. In addition, we establish the minimax rates for estimating normal mean matrices with submatrix sparsity, where the sparsity constraint introduces an additional term in the rate which, in contrast to the unconstrained case, is determined by the smoothness (Lipschitz constant) of the norm. This method is also applicable to the low-rank matrix completion problem and extends well beyond the additive noise model. In particular, it yields tight rates in covariance matrix estimation and Poisson rate matrix estimation problems for all unitarily invariant norms.

Stability of low-rank matrix recovery and its connections to Banach space geometry

There are well-known relationships between compressed sensing and the geometry of the finite-dimensional ℓp spaces. A result of Kashin and Temlyakov [20] can be described as a characterization of the stability of the recovery of sparse vectors via ℓ1-minimization in terms of the Gelfand widths of certain identity mappings between finite-dimensional ℓ1 and ℓ2 spaces, whereas a more recent result of Foucart, Pajor, Rauhut and Ullrich [16] proves an analogous relationship even for ℓp spaces with p<1. In this paper we prove what we call matrix or noncommutative versions of these results: we characterize the stability of low-rank matrix recovery via Schatten p-(quasi-)norm minimization in terms of the Gelfand widths of certain identity mappings between finite-dimensional Schatten p-spaces.

Approximative Compactness and Radon-Nikodym Property inw∗Nearly Dentable Banach Spaces and Applications

Authors definew∗nearly dentable Banach space. Authors study Radon-Nikodym property, approximative compactness and continuity metric projector operator inw∗nearly dentable space. Moreover, authors obtain some examples ofw∗nearly dentable space in Orlicz function spaces. Finally, by the method of geometry of Banach spaces, authors give important applications ofw∗nearly dentability in generalized inverse theory of Banach space.

The non-linear geometry of Banach spaces after Nigel Kalton

This is a survey of some of the results which were obtained in the last twelve years on the non-linear geometry of Banach spaces. We focus on the contribution of the late Nigel Kalton.

Approximative compactness and continuity of the set-valued metric generalized inverse in Banach spaces

In this paper, continuity of the set-valued metric generalized inverse T∂ in approximatively compact Banach spaces is investigated by means of the methods of geometry of Banach spaces. Necessary and sufficient conditions for upper semicontinuity (continuity) for the set-valued metric generalized inverses T∂ are given. Moreover, authors also prove that if X is a nearly dentable space and H is a hyperplane of X, then H is approximatively compact iff PH(x) is compact for any x∈X.

Geometry of Banach spaces with an octahedral norm

We discuss the geometry of Banach spaces whose norm is octahedral or, more generally, locally or weakly octahedral. Our main results characterize these spaces in terms of covering of the unit ball.