Approximation of lipschitz functions by Δ-convex functions in banach spaces

In this paper we give some result about the approximation of a Lipschitz function on a Banach space by means of Δ-convex functions. In particular, we prove that the density of Δ-convex functions in the set of Lipschitz functions for the topology of uniform convergence on bounded sets characterizes the superreflexivity of the Banach space. We also show that Lipschitz functions on superreflexive Banach spaces are uniform limits on the whole space of Δ-convex functions.     

Compact operators in Banach lattices

Disjoint sequence methods from the theory of Riesz spaces are used to study compact operators on Banach lattices. A principal new result of the paper is that each positive map from a Banach latticeE to a Banach latticeF with compact majorant is itself compact provided the norms onE′ andF are order continuous.     

A local Hahn–Banach theorem and its applications

An important consequence of the Hahn–Banach theorem says that on any locally convex Hausdorff topological space X, there are sufficiently many continuous linear functionals to separate points of X. In the paper, we establish a “local” version of this theorem. The result is applied to study the uo-dual of a Banach lattice that was recently introduced in Gao et al. (Positivity 22(3):711–725, 2018). We also provide a simplified approach to the measure-free characterization of uniform integrability established in Kardaras (J Funct Anal 266:1913–1927, 2014).     

Locally Asplund Preduals of Spaces of Holomorphic Functions

Defant [5] introduced the local Radon–Nikodym property for duals of locally convex spaces. This is a generalization of Asplund spaces as defined in Banach space theory. In this paper we generalise Dunford"s Theorem [7] to Banach spaces with Schauder decompositions and apply this result to spaces of holomorphic functions on balanced domains in a Banach space. This revised version was published online in June 2006 with corrections to the Cover Date.     

On universal Banach spaces of density continuum

We consider the question whether there exists a Banach space X of density continuum such that every Banach space of density at most continuum isomorphically embeds into X (called a universal Banach space of density c). It is well known that ℓ_∞/c ₀ is such a space if we assume the continuum hypothesis. Some additional set-theoretic assumption is indeed needed, as we prove in the main result of this paper that it is consistent with the usual axioms of set-theory that there is no universal Banach space of density c. Thus, the problem of the existence of a universal Banach space of density c is undecidable using the usual axioms of set-theory.     

Stability Analysis for Minty Mixed Variational Inequality in Reflexive Banach Spaces

This paper is devoted to the stability analysis for a class of Minty mixed variational inequalities in reflexive Banach spaces, when both the mapping and the constraint set are perturbed. Several equivalent characterizations are given for the Minty mixed variational inequality to have nonempty and bounded solution set. A stability result is presented for the Minty mixed variational inequality with Φ-pseudomonotone mapping in reflexive Banach space, when both the mapping and the constraint set are perturbed by different parameters. As an application, a stability result for a generalized mixed variational inequality is also obtained. The results presented in this paper generalize and extend some known results in Fan and Zhong (Nonlinear Anal., Theory Methods Appl. 69:2566–2574, 2008) and He (J. Math. Anal. Appl. 330:352–363, 2007).     

Viscosity approximation methods for resolvents of accretive operators in Banach spaces

In this paper, we first prove a strong convergence theorem for resolvents of accretive operators in a Banach space by the viscosity approximation method, which is a generalization of the results of Reich [J. Math. Anal. Appl. 75 (1980), 287–292], and Takahashi and Ueda [J. Math. Anal. Appl. 104 (1984), 546–553]. Further using this result, we consider the proximal point algorithm in a Banach space by the viscosity approximation method, and obtain a strong convergence theorem which is a generalization of the result of Kamimura and Takahashi [Set-Valued Anal. 8 (2000), 361–374]. Dedicated to the memory of Jean Leray     

On the weak compactness in Banach lattices

The Grothendieck's criterion of weak compactness in spaceC(S) has been extended by C.P.Niculescu for weakly sequentially complete Banach lattices. This paper first provides some sufficient and necessary conditions for weakly sequentially complete Banach lattices, with the result that the criterion C.P.Niculescu obtained of the weak compactness has actually characterized the weakly sequentially Banach lattices. At the same time we extend C.P.Niculescu's important results. Then we solve negatively Problem 1.11 which was posed by C.P.Niculescu, and a sufficient condition for this is given which generalizes Pelczyński's result.     

Characterizations for solidness of dual cones with applications

In this paper, some characterizations for the solidness of dual cones are established. As applications, we prove that a Banach space is reflexive if it contains a solid pointed closed convex cone having a weakly compact base, and prove an analogue of a Karamardian’s result for the linear complementarity problem in reflexive Banach spaces. The uniqueness of the solution of the linear complementarity problem is also discussed.     

On product formulas for exponential functions

A direct and inverse theorem on the generalized Chernoff formula in Banach algebras is proved by using only elementary properties of exponential and logarithm functions. The main result in this paper forms a part of the first author’s Master thesis (1981) at National Central University.     

Gaussian Chaos Laws on Banach Function Spaces

We give a characterization of Gaussian chaos laws on Banach function spaces which do not contain ℓ _∞ ⁿ 's uniformly. The result is applied to describe the convergence in law of U-processes with sample paths in certain Banach function spaces. __________ Published in Lietuvos Matematikos Rinkinys, Vol. 45, No. 4, pp. 553–566, October–December, 2005.     

Existence of equilibrium for economies with infinitely many commodities and infinitely countable consumers

This paper examines the existence of equilibria for double infinite eonomies. S.F. Richard and S. Srivastava have established the existence of equilibria for economies with infinitely countable consumers when commodity space isL ^∞ (M, M, μ). However, most Banach Lattices as commodity spaces haven’t interior points in their positive cones, so their result can’t be applied to many cases. We here consider a general Banach Lattice as commodity space and introduce a concept of equiproperness on preferences. Under the assumption the existence of equilibrium for economy is established. Finally, we examine the existence of equilibria for production economies.     

Perturbation of operators and applications to frame theory

A celebrated classical result states that an operator U on a Banach space is invertible if it is close enough to the identity operator I in the sense that ‖I−U‖<1. Here we show that U actually is invertible under a much weaker condition. As an application we prove new theorems concerning stability offrames (and frame-like decompositions) under perturbation in both Hilbert spaces and Banach spaces. The first named author is partially supported by grants from the U.S. National Science Foundation (grant no. NSF DMS-9201357), the Danish Natural Science Research Council (Grant no. 9401598), and grants from the University of Missouri System Research Board, and the MU Research Council. The second named author thanks the University of Missouri for its hospitality during a visit, where the first draft of the paper was written.     

Weakly stable Banach spaces

The class of stable Banach spaces, inspired by the stability theory in mathematical logic, was introduced by Krivine and Maurey and provided the proper context for the abstract formulation of Aldous’ result of subspaces ofL ¹. In this paper we study the wider class of weakly stable Banach spaces, where the exchangeability of the iterated limits occurs only for sequences belonging to weakly compact subsets, introduced independently by Garling (in an earlier unpublished version of his expository paper on stable Banach spaces brought recently to our attention) and by the authors. Taking into account Rosenthal’s application of the study of pointwise compact sets of Baire-1 functions (Rosenthal compact spaces) in the study of Banach spaces (for whichl ¹ does not embed isomorphically) and of the study of Rosenthal compact sets by Rosenthal and Bourgain-Fremlin-Talagrand, we prove the following analogue of the Krivine-Maurey theorem for weakly stable spaces:If X is infinite dimensional and weakly stable then either l ^p for some p≧1or c_o embeds isomorphically in X (§1). Garling (in the above reference) proved this result under the additional assumption thatX* is separable. We also construct an example of a Banach spaceX which is weakly stable, without an equivalent stable norm, and such thatl ² embeds isomorphically in every infinite dimensional subspace ofX (§3).     

Elementary rotations of operators on regular Banach spaces

In this paper, we prove the existence of an elementary rotation (a Julia operator) for any continuous, linear, adjointable operator in a regular Banach space with inner product. The proof is based on a more general theorem of the author on the existence of an elementary rotation for any linear operator in a category with quadratic splitting. This result is a generalization of a well-known result about the existence of an elementary rotation for any continuous linear operator in a Krein space. The result can be useful for constructing isometric and unitary dilations as well as characteristic functions of continuous linear operators acting in regular Banach spaces with inner product. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 6, pp. 175–192, 2006.     

M-structure in dual Banach spaces

A result previously known only for certain ordered Banach spaces is generalized to arbitrary real Banach spaces. Let ℒ be the Banach algebra of operators generated by theL-projections of a real Banach spaceU, and let ℳ (U ^* be the bounded operators on the dual spaceU ^* with adjoint in ℒ(U ^**. Then the adjoint operation maps ℒ (U) onto ℳ (U ^*). In particular, anyM-projection ofU ^* is weak^* continuous. Supported in part by the National Science Foundation.     

Embedding unconditional stable banach spaces into symmetric stable banach spaces

In this paper we prove the following result which solves a question raised by A. Pelczynski: “Every stable Banach space with an unconditional basis is isomorphic to a complemented subspace of some stable Banach space with a symmetric basis.” Moreover, we show that all the interpolation spacesl ^p ,l ^q _θ,X,1 1≦p, q<∞ andX stable, are stable.     

A substitute for the Sard-Smale theorem in theC 1 case

IfF is a Fredholm mapping of indexΝ ∃ ℤ and classC ^max(Ν,0)+1 between separable Banach spaces, the Sard—Smale theorem yields the existence of arbitrarily small perturbations ofF having 0 as a regular value. The smoothness requirement cannot be weakened in the Sard—Smale theorem itself, at least whenΝ ≥ 0, but we prove that the approximation result remains valid irrespective of the indexΝ whenF is only of classC ¹ and satisfies appropriate properness-like conditions. The separability of the spaces is not needed either. Everything carries over to the setting of Banach manifolds modeled on spaces with a norm of classC ¹ away from the origin. We also show that in Banach spaces, theC ¹ norm assumption can be dropped without major prejudice. The application to degree theory forC ¹ Fredholm mappings of index 0 is developed in a separate paper.     

On quasi-convex mappings of order α in the unit ball of a complex Banach space

In this paper, a class of biholomorphic mappings named quasi-convex mapping of order α in the unit ball of a complex Banach space is introduced. When the Banach space is confined to ℂ ⁿ , we obtain the relation between this class of mappings and the convex mappings. Furthermore, the growth and covering theorems of this class of mappings are given on the unit ball of a complex Banach space X. Finally, we get the second order terms coefficient estimations of the homogeneous expansion of quasi-convex mapping of order α defined on the polydisc in ℂ ⁿ and on the unit ball in a complex Banach space, respectively. Dedicated to Professor Sheng GONG on the occasion of his 75th birthday     

Irreflexive Banach spaces are imperfect

It is shown that irreflexive Banach spaces are imperfect: thus the “perfect” spaces are exactly the reflexive Banach spaces and “mixed” spaces do not exist. This paper is part of the author’s M.Sc. dissertation, performed under guidance of Dr. D. Amir at the Technion — Israel Institute of Technology. The author wishes to thank Dr. Amir for his help.