On Approximation of Measures by Their Finite-Dimensional Images

Functional Analysis and Its Applications - We consider Borel measures on separable Banach spaces that are limits of their finite-dimensional images in the weak topology. The class of Banach spaces...     

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.     

Some estimations of Banach limits

In the first part of this paper, we introduce the notions of upper weight, lower weight and weight of subsequences of natural numbers and investigate some new estimations about Banach limits by using some results from Sucheston [L. Sucheston, On existence of finite invariant measures, Math. Z. 86 (1964) 327-336; L. Sucheston, Banach limit, Amer. Math. Monthly 74 (1967) 308-311]. In the second part of this paper, we study the connections between weights and densities of subsequences of natural numbers, and give a familiar formula to find some values of Banach limits on almost convergent sequences.     

On vector-valued Banach limits

In this brief communication we propose a vector-valued version of Lorentz’ intrinsic characterization of almost convergence, for which we find a legitimate extension of the concept of Banach limit to vector-valued sequences. Banach spaces 1-complemented in their biduals admit vector-valued Banach limits, whereas c ₀ does not.     

On a Type of Fréchet Principal Bundles over Banach Bases

In this paper we study a certain class of Fréchet principal bundles. Those which have structural groups obtained as projective limits of Banach Lie groups. In particular, we prove that each bundle of the previous type can be thought of as a projective limit of Banach principal bundles and any connection of them is a generalized limit of Banach connections. Using the previous, we achieve to translate in the Fréchet case basic geometric properties known so far only for Banach bundles.     

Generalized measures of noncompactness of sets and operators in Banach spaces

New measures of noncompactness for bounded sets and linear operators, in the setting of abstract measures and generalized limits, are constructed. A quantitative version of a classical criterion for compactness of bounded sets in Banach spaces by R. S. Phillips is provided. Properties of those measures are established and it is shown that they are equivalent to the classical measures of noncompactness. Applications to summable families of Banach spaces, interpolations of operators and some consequences are also given.     

Projective limits of weighted (LB)-spaces of continuous functions

Countable projective spectra of countable inductive limits, called (PLB)-spaces, of weighted Banach spaces of continuous functions are investigated. It is characterized when the derived projective limit functor vanishes in terms of the sequences of the weights defining the spaces. The locally convex properties of the corresponding projective limits are analyzed, too. Received: 30 January 2009     

Functions of shift operator and their applications to difference equations

We study the representation for functions of shift operator acting upon bounded sequences of elements of a Banach space. An estimate is obtained for the bounded solution of a linear difference equation in the Banach space. For two types of differential equations in Banach spaces, we present sufficient conditions for their bounded solutions to be limits of bounded solutions of the corresponding difference equations and establish estimates for the rate of convergence.     

Preduals of Campanato Spaces and Sobolev-Campanato Spaces: A General Construction

In this paper we describe two limiting processes for families of Banach spaces closely related to the standard definition of projective and inductive limits. These processes lead again to Banach spaces. Information about linear operators and duality between basic families of spaces is carried over to the corresponding limit spaces. The abstract results are shown to be applicable to Campanato spaces and Sobolev-Campanato spaces. In particular, we obtain the existence and a characterization of predual spaces. Some imbedding relations are investigated in more detail.     

Extensions of Banach Lie-Poisson spaces

The extension of Banach Lie-Poisson spaces is studied and linked to the extension of a special class of Banach Lie algebras. The case of W∗-algebras is given particular attention. Semidirect products and the extension of the restricted Banach Lie-Poisson space by the Banach Lie-Poisson space of compact operators are given as examples.     

Banach 空间中一阶周期边值问题的广义解

In this paper, the authors study the periodic boundary value problems of a class of nonlinear integro-differential equations of mixed type in Banach space with Carath′eodory's conditions. We arrive at the conclusion of the existence of generalized solutions between general- ized upper and lower solutions, and develop the monotone iterative technique to find generalized extremal solutions as limits of monotone solution sequences in Banach space.     

Generalized Limits and Statistical Convergence

Consider the Banach space m of real bounded sequences, x, with \({\Vert x\Vert =\sup_{k}|x_{k}|}\). A positive linear functional L on m is called an S-limit if \({L(\chi _{K})=0}\) for every characteristic sequence \({\chi _{K} }\) of sets, K, of natural density zero. We provide regular sublinear functionals that both generate as well as dominate S-limits. The paper also shows that the set of S-limits and the collection of Banach limits are distinct but their intersection is not empty. Furthermore, we show that the generalized limits generated by translative regular methods is equal to the set of Banach limits. Some applications are also provided.     

On quantification of weak sequential completeness

We consider several quantities related to weak sequential completeness of a Banach space and prove some of their properties in general and in L-embedded Banach spaces, improving in particular an inequality of G. Godefroy, N. Kalton and D. Li. We show some examples witnessing natural limits of our positive results, in particular, we construct a separable Banach space X with the Schur property that cannot be renormed to have a certain quantitative form of weak sequential completeness, thus providing a partial answer to a question of G. Godefroy.     

Affine mappings of invertible operators

The infinite-dimensional analogues of the classical general linear group appear as groups of invertible elements of Banach algebras. Mappings of these groups onto themselves that extend to affine mappings of the ambient Banach algebra are shown to be linear exactly when the Banach algebra is semi-simple. The form of such linear mappings is studied when the Banach algebra is a C*-algebra.

     

PLB-spaces of holomorphic functions with logarithmic growth conditions

Countable projective limits of countable inductive limits, so-called PLB-spaces, of weighted Banach spaces of continuous functions have recently been investigated by Agethen, Bierstedt and Bonet. In a previous article, the author extended their investigation to the case of holomorphic functions and characterized when spaces over the unit disc w.r.t. weights of polynomial decay are ultrabornological or barrelled. In this note, we prove a similar characterization for the case of weights which tend to zero logarithmically.     

Representation of certain infinitely divisible probability measures on Banach spaces

Subclasses L₀ ? L₁ ? … ? L_∞ of the class L₀ of self-decomposable probability measures on a Banach space are defined by means of certain stability conditions. Each of these classes is closed under translation, convolution and passage to weak limits. These subclasses are analogous to those defined earlier by K. Urbanik on the real line and studied in that context by him and by the authors. A representation is given for the characteristic functionals of the measures in each of these classes on conjugate Banach spaces. On a Hilbert space it is shown that L_∞ is the smallest subclass of L₀ with the closure properties above containing all the stable measures.     

PBVP for integro-differential equations of volterra type in B.S.

In this paper, we study the PBVP for integro-differential equations of Volterra type in Banach spaces. By developing monotone iterative technique for the PBVP, we get some results concerning the existence of extremal solutions, which are the limits of monotone sequences.The project supported by the Natural Science Foundation of Shandong Province.     

On Fredholm index in Banach spaces

We prove the stability of the index of a Fredholm complex of Banach spaces under those compact perturbations which are uniform limits of finite-rank operators (Theorem 3.3). This result is a consequence of some similar statements (Theorems 3.1 and 3.2) concerning more general objects, namely the Fredholm pairs (Definition 1.1).     

Vector valued Banach limits and generalizations applied to the inhomogeneous Cauchy equation

Aequationes mathematicae - In Prager and Schwaiger (Grazer Math Ber 363:171–178, 2015) the classical notion of Banach limits was used to solve the inhomogeneous Cauchy equation...     

Nonsmooth sequential analysis in Asplund spaces

We develop a generalized differentiation theory for nonsmooth functions and sets with nonsmooth boundaries defined in Asplund spaces. This broad subclass of Banach spaces provides a convenient framework for many important applications to optimization, sensitivity, variational inequalities, etc. Our basic normal and subdifferential constructions are related to sequential weak-star limits of Fréchet normals and subdifferentials. Using a variational approach, we establish a rich calculus for these nonconvex limiting objects which turn out to be minimal among other set-valued differential constructions with natural properties. The results obtained provide new developments in infinite dimensional nonsmooth analysis and have useful applications to optimization and the geometry of Banach spaces.