Local Lipschitz-α Mappings and Applications to Sublinear Expectations

The aim of this paper is to establish a series of important properties of local Lipschitz-α mappings from a subset of a normed space into a normed space. These mappings include Lipschitz operators, Lipschitz-α operators and local Lipschitz functions. Some applications to the theory of sublinear expectation spaces are given.     

Best approximation by polynomial operators in banach spaces

The problem of approximation in the space of bounded linear operators ? (E;G) between normed spaces E and G by compact operators has been extensively studied in the last few years.

Recently Deutsch, Mach and Saatkamp ([2]) have considered the problem of approximating elements of ?(E;G) by the subset K _N(E;G) of operators whose range is at most N dimensional. We consider in this paper the problem of approximating operators (not necessarily linear) beteen normed spaces E and G by continuous homogeneous polynomials, and in particular by such polynomials which have finite-dimensional range.     

On The Spaces of Linear Operators Acting Between Asymmetric Cone Normed Spaces

An asymmetric norm is a positive sublinear functional p on a real vector space X satisfying \(x=\theta _X\) whenever \(p(x)=p(-x)=0\). Since the space of all lower semi-continuous linear functionals of an asymmetric normed space is not a linear space, the theory is different in the asymmetric case. The main purpose of this study is to define bounded and continuous linear operators acting between asymmetric cone normed spaces. After examining the differences with symmetric case, we give some results related to Baire’s characterization of completeness in asymmetric cone normed spaces.     

Paracomplete normed spaces and Fredholm theory

A normed space is paracomplete if it admits a new norm, stronger than the initial one, that makes it complete. Here we give a characterization of paracomplete normed spaces. As a consequence, we show that operators on paracomplete spaces have compact spectrum in the algebra of all operators, and that the class of paracomplete spaces is not stable under ℓ₂-sums. Moreover, we give characterizations for the closed Fredholm operators on paracomplete spaces and for the almost semi-Fredholm operators of Harte on normed spaces.     

关于Z-空间的对偶空间的讨论

本文以自然的方式定义了从Z-空间X到Z-空间Y的有界线性算子的和以及它们的数乘.从而得到了与赋范空间的对偶空间理论类似的一系列结论.     

A unified framework for the discretization of nonlinear operator equations

A general theory for the discretization of non-linear operator equations is presented. A given operator with certain continuity and compactness properties is approximated by a sequence of operators acting in different spaces, usually finite dimensional. Connection maps, such as restriction and interpolation, relate the spaces. The abstract convergence theory is formulated in terms of metric spaces. Specializations and applications to differential and integral equations involve normed linear spaces. The case with the same setting for the original and approximate problems was treated in [1]. For typical problems, both types of discretization methods are available. They are related by means of the connection maps.     

Topological degree theory and fixed point theorems in fuzzy normed space

In this paper, the Leray–Schauder topological degree theory is developed in a fuzzy normed space. Since the linear topology on this fuzzy normed space is not necessarily locally convex, and since each Menger probabilistic normed space can be considered as a special fuzzy normed space, the degree theory in this paper is different from the degree theory in locally convex linear topological space presented by Nagumo (Amer. J. Math. 73 (1951) 497–511), and it also is an extension of the degree theory in Menger probabilistic normed space studied by Zhang and Chen (Appl. Math. Mech. 10(6) (1989) 477–486). Applying this degree theory, some fixed point theorems for operators are given in fuzzy normed spaces, and some former corresponding results are extended and improved.     

Invariance of the order and type of a sequence of operators

In the paper, the invariance property of characteristics (the order and type) of an operator and of a sequence of operators with respect to a topological isomorphism is proved. These characteristics give precise upper and lower bounds for the expressions ‖A_n(x)‖_p and enable one to state and solve problems of operator theory in locally convex spaces in a general setting. Examples of such problems are given by the completeness problem for the set of values of a vector function in a locally convex space, the structure problem for a subspace invariant with respect to an operator A, the problem of applicability of an operator series to a locally convex space, the theory of holomorphic operator-valued functions, the theory of operator and differential-operator equations in nonnormed spaces, and so on. However, the immediate evaluation of characteristics of operators (and of sequences of operators) directly by definition is practically unrealizable in spaces with more complicated structure than that of countably normed spaces, due to the absence of an explicit form of seminorms or to their complicated structure. The approach that we use enables us to find characteristics of operators and sequences of operators using the passage to the dual space, by-passing the definition, and makes it possible to obtain bounds for the expressions ‖A_n(x)‖_p even if an explicit form of seminorms is unknown.     

Some remarks on Birkhoff–James orthogonality of linear operators

We study Birkhoff–James orthogonality of compact (bounded) linear operators between Hilbert spaces and Banach spaces. Applying the notion of semi-inner-products in normed linear spaces and some related geometric ideas, we generalize and improve some of the recent results in this context. In particular, we obtain a characterization of Euclidean spaces and also prove that it is possible to retrieve the norm of a compact (bounded) linear operator (functional) in terms of its Birkhoff–James orthogonality set. We also present some best approximation type results in the space of bounded linear operators.     

Weakly compact operators on non-complete normed spaces

We prove that weakly compact operators on a non-reflexive normed space cannot be bijective. We also show that, in the above result, bijectivity cannot be relaxed to surjectivity. Finally, we study the behaviour of surjective weakly compact operators on a non-reflexive normed space, when they are perturbed by small scalar multiples of the identity, and derive from this study the recent result of Spurný [A note on compact operators on normed linear spaces, Expo. Math. 25 (2007) 261–263] that compact operators on an infinite-dimensional normed space cannot be surjective.     

Convolution operators on spaces of entire functions

We show that nontrivial convolution operators on certain spaces of entire functions on E are frequently hypercyclic when E is a normed space and when E is the strong dual of a Fréchet nuclear space. We also obtain results of existence and approximation for convolution equations on certain spaces of entire functions on arbitrary locally convex spaces.     

关于随机共轭空间的各种定义及随机线性泛函各种有界性的某些评论

中心目的是详细廉政论在随机共轭空间理论形成过程中所经历的三个阶段的工作，尤其指出了这三个阶段工作之间的联系及本质差别；给出了强有界、拓扑有界及几乎处处有界随机线性泛函之间的关系；亦指出了在概率赋范空间上线性算子理论研究中目前存在的不足．     

Normes des extensions quaternionique d'opérateurs réels

We consider bounded linear operators defined on real normed spaces, and with range in quaternionic spaces. We study the norms of the quaternionic extensions of such operators. To cite this article: D. Alpay et al., C. R. Acad. Sci. Paris, Ser. I 340 (2005).     

随机赋范空间算子随机范数与Hahn-Banach延拓定理

基于概率测度理论基础,研究了随机赋范空间中算子随机范数,得到了线性算子空间与线性泛函的若干随机化结果与随机化的Hahn-Banach延拓定理.结果可能成为随机泛函分析与概率论及应用的理论工具.     

Best coapproximation in normed linear spaces

We consider, in normed linear spaces, a kind of approximation by elements of linear subspaces, introduced byC. Franchetti andM. Furi [5], which we call best coapproximation. We obtain some results on characterization and existence of elements of best coapproximation in arbitrary normed linear spaces and in spaces of continuous functions. We give some characterizations of strict convexity in terms of best coapproximation and we study some properties of the setvalued operators of best coapproximation.Work performed partially under the auspices of the GNAFA (National Group for Functional Analysis and its Applications) of the CNR (National Research Council of Italy)     

Fredholm theory forp-adic locally convex spaces

Summary The authors develop the Fredholm theory for semi-compact operators on non-archimedean locally convex spaces. This theory coincides with Schikhof's Fredholm theory for compact operators on Banach spaces which fails for non-complete normed spaces.     

Spectral measures,III: Densely defined spectral measures

In this paper we extend the theory of spectral measures developed in Parts I and II to the case where values are assumed in the set of discontinuous (in normed spaces „unbounded”) operators. Examples of operators in nonlocally convex spaces are given, which have densely defined measures.     

Singular operators and Fourier multipliers in weighted Lebesgue spaces with variable index

Mikhlin’s ideas and results related to the theory of spaces L _ρ ^p(·) with nonstandard growth are developed. These spaces are called Lebesgue spaces with variable index; they are used in mechanics, the theory of differential equations, and variational problems. The boundedness of Fourier multipliers and singular operators on the spaces L _ρ ^p(·) are considered. All theorems are derived from an extrapolation theorem due to Rubio de Francia. The considerations essentially use theorems on the boundedness of operators and maximal Hardy-Littlewood functions on Lebesgue spaces with constant index.     

NOTE ON SOME CHARACTERISATIONS OF UNBOUNDED WEAKLY COMPACT OPERATORS

Abstract

Some well known properties of bounded weakly compact operators in Banach spaces are shown to be valid for arbitrary operators in normed spaces.     

概率度量空间中的拓朴度理论与不动点定理*

在本文中我们在概率线性赋范空间中建立了Leray-Schauder度理论.并以此为工具得出了概率线性赋范空间中的某些不动点定理.