Structure Of Banach Spaces Research Articles

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.

Approximation of functions in variable-exponent Lebesgue and Sobolev spaces by de la Vallée-Poussin means

We consider the space formed by -periodic real measurable functions for which the integral exists and is finite, where , , is a -periodic measurable function (a variable exponent). If , then the space can be endowed with the structure of Banach space with the norm 0: \\int_{-\\pi}^{\\pi}\\biggl|\\frac{f(x)}{\\alpha}\\biggr|^{p(x)}\\,dx\\le1\\biggr\\}. \\end{equation*} ?> In the space we distinguish a subspace of Sobolev type. We investigate the approximation properties of the de la Vallée-Poussin means for trigonometric Fourier sums for functions in the space . In particular, we prove that if the variable exponent satisfies the Dini-Lipschitz condition and if , then the de la Vallée-Poussin means with satisfy where is a modulus of continuity of the function defined in terms of the Steklov functions. It is proved that if , , and the Dini-Lipschitz condition holds, then where stands for the best approximation to by trigonometric polynomials of order . Bibliography: 19 titles.

Asymptotically hilbertian modular Banach spaces: examples of uncountable categoricity

We give a criterion ensuring that the elementary class of a modular Banach space \(E\) (that is, the class of Banach spaces, some ultrapower of which is linearly isometric to an ultrapower of \(E\)) consists of all direct sums \(E\oplus_m H\), where \(H\) is an arbitrary Hilbert space and \(\oplus_m\) denotes the modular direct sum. Also, we give several families of examples in the class of Nakano direct sums of finite dimensional normed spaces that satisfy this criterion. This yields many new examples of uncountably categorical Banach spaces, in the model theory of Banach space structures.

Banach space structure of weighted Fock-Sobolev spaces

We discuss the Banach space structure of the fractional order weighted Fock-Sobolev spaces F , , mainly include giving some growth estimates for Fock-Sobolev functions and approximating them by a sequence of ‘nice’ functions in two different ways.

On the structure of Lipschitz-free spaces

In this note we study the structure of Lipschitz-free Banach spaces. We show that every Lipschitz-free Banach space over an infinite metric space contains a complemented copy of ℓ 1 \ell _1 . This result has many consequences for the structure of Lipschitz-free Banach spaces. Moreover, we give an example of a countable compact metric space K K such that F ( K ) \mathcal {F}(K) is not isomorphic to a subspace of L 1 L_1 and we show that whenever M M is a subset of R n \mathbb {R}^n , then F ( M ) \mathcal {F}(M) is weakly sequentially complete; in particular, c 0 c_0 does not embed into F ( M ) \mathcal {F}(M) .

(β)-distortion of some infinite graphs

A distortion lower bound of $\Omega(\log(h)^{1/p})$ is proven for embedding the complete countably branching hyperbolic tree of height $h$ into a Banach space admitting an equivalent norm satisfying property $(\beta)$ of Rolewicz with modulus of power type $p\in(1,\infty)$ (in short property ($\beta_p$)). Also it is shown that a distortion lower bound of $\Omega(\ell^{1/p})$ is incurred when embedding the parasol graph with $\ell$ levels into a Banach space with an equivalent norm with property ($\beta_p$). The tightness of the lower bound for trees is shown adjusting a construction of Matou\v{s}ek to the case of infinite trees. It is also explained how our work unifies and extends a series of results about the stability under nonlinear quotients of the asymptotic structure of infinite-dimensional Banach spaces. Finally two other applications regarding metric characterizations of asymptotic properties of Banach spaces, and the finite determinacy of bi-Lipschitz embeddability problems are discussed.

Dedekind Complete Posets from Sheaves on von Neumann Algebras

We show that for any two von Neumann algebras M and N, the space of non-unital normal homomorphisms N→M with finite support, modulo conjugation by unitaries in M, is Dedekind complete with respect to the partial order coming from the addition of homomorphisms with orthogonal ranges; this ties in with work by Brown and Capraro, where the corresponding objects are given Banach space structures under various niceness conditions on M and N. More generally, we associate to M a Grothendieck site-type category, and show that presheaves satisfying a sheaf-like condition on this category give rise to Dedekind complete lattices upon modding out unitary conjugation. Examples include the above-mentioned spaces of morphisms, as well as analogous spaces of completely positive or contractive maps. We also study conditions under which these posets can be endowed with cone structures extending their partial additions.

Some results in fixed point theory and application to the convergence of some iterative processes

AbstractIn this paper, we study the existence and uniqueness of fixed points for a class of self-mappings satisfying certain rational expressions on closed, bounded and convex subsets with normal structures in reflexive Banach spaces. We show that, in particular, this class extends that introduced by Ray and Singh (Indian J. Pure Appl. Math. 9:216-221, 1978). As an application, we give an investigation of the convergence and stability of some iterative processes associated to these mappings.

Optimality Conditions for Disjunctive Optimization in Reflexive Banach Spaces

We study optimization problems with the constraints having disjunctive structures in reflexive Banach spaces. By the representations of contingent cones and Frechet normal cones to finite unions of sets in general Banach spaces and using the special structures of convex generalized polyhedral sets, we calculate the Mordukhovich normal cones to finite unions of closed and convex sets that particularly include convex generalized polyhedral sets in reflexive Banach spaces. Furthermore, based on these calculations and the Guignard-type constraint qualifications, we derive new optimality conditions for disjunctive optimization problems. We also present specializations of these results to optimization problems with variational inequality constraints.

Geometry of Banach Spaces, Operator Theory, and Their Applications

In recent years, with developments of computer, high technique, and life science, more and more requirements were proposed to functional analysis. There were a large number of research articles published in the relative area and many major discoveries have been obtained. At the present time the ideas, terminology, and methods of functional analysis have penetrated deeply, not only into natural sciences, but also into applied disciplines. Many real world problems were analyzed and solved by using the ideas and results of function analysis. To fit the requirements, we edit this special issue. The special issue represented the recent development in geometric structure of Banach spaces, Bergman spaces, and Minkowski spaces, random normed modules on function space, linear and nonlinear operator theory, and the applications of modern analysis in related areas of mathematics, as well as other disciplines, such as economics, finance and risk management, dynamic systems, the natural and life sciences, medicine, physics, and other real world problems. The special issue consisted of twenty research articles contributed by participants of the 4th International Conference on Modern Analysis and Its Applications which was held at the Tianjin University, China, from August 1 to 4, 2013. With over 240 participants fromdifferent countries, the conference proved highly successful in bringing together experts and researchers and provided an opportunity to evaluate and disseminate new ideas and methods in modern analysis. Each paper has been peer-reviewed and the editors thank the referees for their time and efforts to help ensure a high standard for this special issue. We hope that readers interested in function spaces and operator theory will find in this special issue not only new information about development of functional analysis, but also important questions be resolved and directions for further study. At same time, we hope that in the near future, we can see new results to be published based on this special issue.

Existence and Multiple Positive Solutions for Boundary Value Problem of Fractional Differential Equation withp-Laplacian Operator

This paper investigates the existence, multiplicity, nonexistence, and uniqueness of positive solutions to a kind of two-point boundary value problem for nonlinear fractional differential equations withp-Laplacian operator. By using fixed point techniques combining with partially ordered structure of Banach space, we establish some criteria for existence and uniqueness of positive solution of fractional differential equations withp-Laplacian operator in terms of different value of parameter. In particular, the dependence of positive solution on the parameter was obtained. Finally, several illustrative examples are given to support the obtained new results. The study of illustrative examples shows that the obtained results are applicable.

Existence of a positive solution for one-dimensional singular p-Laplacian problems and its parameter dependence

We consider one-dimensional singular p-Laplacian problems of the form{λ(φp(u′))′+ω(t)f(t,u)=0,t∈(0,1),au(0)−bu′(0)=∫01g(t)u(t)dt,u′(1)=0, where λ is a positive parameter, φp(s)=|s|p−2s, p>1, (φp)−1=φq, 1p+1q=1, and ω may be singular at t=0 and/or t=1. Using fixed-point techniques combined with the partially ordered structure of Banach space, we establish some new and more general existence, multiplicity, and nonexistence results. We also study the dependence of the positive solution uλ(t) on the parameter λ, that is,limλ→+∞‖uλ‖=+∞orlimλ→+∞‖uλ‖=0. An example illustrates our main results.

New results on q-positivity

In this paper we discuss symmetrically self-dual spaces, which are simply real vector spaces with a symmetric bilinear form. Certain subsets of the space will be called q-positive, where q is the quadratic form induced by the original bilinear form. The notion of q-positivity generalizes the classical notion of the monotonicity of a subset of a product of a Banach space and its dual. Maximal q-positivity then generalizes maximal monotonicity. We discuss concepts generalizing the representations of monotone sets by convex functions, as well as the number of maximally q -positive extensions of a q-positive set. We also discuss symmetrically self-dual Banach spaces, in which we add a Banach space structure, giving new characterizations of maximal q-positivity. The paper finishes with two new examples.

Jordan structures in Banach spaces

Jordan structures in Banach spaces

The uniform structure of Banach spaces

We explore the existence of uniformly continuous sections for quotient maps. Using this approach we are able to give a number of new examples in the theory of the uniform structure of Banach spaces. We show for example that there are two non-isomorphic separable \({\mathcal L_1}\)-subspaces of l1 which are uniformly homeomorphic. We also prove the existence of two coarsely homeomorphic Banach spaces (i.e. with Lipschitz isomorphic nets) which are not uniformly homeomorphic (answering a question of Johnson, Lindenstrauss and Schechtman). We construct a closed subspace of L1 whose unit ball is not an absolute uniform retract (answering a question of the author).

RANDOMIZED SERIES AND GEOMETRY OF BANACH SPACES

We study some properties of the randomized series and their applications to the geometric structure of Banach spaces. For $n \geq 2$ and $1 \lt p \lt \infty$, it is shown that $\ell^n_{\infty}$ is representable in a Banach space $X$ if and only if it is representable in the Lebesgue-Bochner $L_p(X)$. New criteria for various convexity properties in Banach spaces are also studied. It is proved that a Banach lattice $E$ is uniformly monotone if and only if its $p$-convexification $E^{(p)}$ is uniformly convex and that a Köthe function space $E$ is upper locally uniformly monotone if and only if its $p$-convexification $E^{(p)}$ is midpoint locally uniformly convex.

Rademacher series and isomorphisms of rearrangement invariant spaces on the finite interval and on the semi-axis

Let X be a rearrangement invariant function space on [ 0 , 1 ] . We consider the subspace Radi X of X which consists of all functions of the form f = ∑ k = 1 ∞ x k r k , where x k are arbitrary independent functions from X and r k are usual Rademacher functions independent of { x k } . We prove that Radi X is complemented in X if and only if both X and its Köthe dual space X ′ possess the so-called Kruglov property. As a consequence we show that the last conditions guarantee that X is isomorphic to some rearrangement invariant function space on [ 0 , ∞ ) . This strengthens earlier results derived in different approach in [W.B. Johnson, B. Maurey, G. Schechtman, L. Tzafriri, Symmetric structures in Banach spaces, Mem. Amer. Math. Soc. 1 (217) (1979)].

Multiplication Operators on the Lipschitz Space of a Tree

In this paper, we study the multiplication operators on the space of complex-valued functions f on the set of vertices of a rooted infinite tree T which are Lipschitz when regarded as maps between metric spaces. The metric structure on T is induced by the distance function that counts the number of edges of the unique path connecting pairs of vertices, while the metric on ℂ is Euclidean. After observing that the space \({\mathcal{L}}\) of such functions can be endowed with a Banach space structure, we characterize the multiplication operators on \({\mathcal{L}}\) that are bounded, bounded below, and compact. In addition, we establish estimates on the operator norm and on the essential norm, and determine the spectrum. We then prove that the only isometric multiplication operators on \({\mathcal{L}}\) are the operators whose symbol is a constant of modulus one. We also study the multiplication operators on a separable subspace of \({\mathcal{L}}\) we call the little Lipschitz space.

Continuous first order logic and local stability

We develop continuous first order logic, a variant of the logic described in \cite{Chang-Keisler:ContinuousModelTheory}. We show that this logic has the same power of expression as the framework of open Hausdorff cats, and as such extends Henson's logic for Banach space structures. We conclude with the development of local stability, for which this logic is particularly well-suited.

An amalgamation of the Banach spaces associated with James and Schreier, Part I: Banach-space structure

An amalgamation of the Banach spaces associated with James and Schreier, Part I: Banach-space structure