The generalized Roper-Suffridge extension operator in Banach spaces (II)

In this paper, we generalize the Roper-Suffridge extension operator from Cn to Banach spaces. It is proved that this operator preserves the biholomorphic ? starlikeness on some domains in Banach spaces. From these, we may construct a lots of concrete examples about biholomorphic ? starlike mappings on some domains Ω in Cn, or Hilbert spaces, or Banach spaces from univalent ? starlike functions on the unit disc U in C. Meanwhile, the growth theorems of the corresponding mappings are given. Some results of Gong and Liu, Roper and Suffridge, Graham et al. in Cn are extended to Hilbert spaces or Banach spaces.     

Holomorphic automorphisms of complex Banach spaces

We consider holomorphic automorphisms of infinite dimensional complex Banach spaces. First we look at automorphisms with an attracting fixed point to construct Fatou–Bieberbach domains in Banach spaces. Second, we look tame sets in Banach spaces. Recall that a discrete set in X is tame if it can be mapped onto an arithmetic progression via an automorphism of X. We show that bounded discrete sets of Banach spaces allowing a Schauder basis are tame. In contrast, \(l_\infty \) has several bounded discrete sets which are not tame.     

在Banach空间中推广的Roper-Suffridge算子(Ⅰ)

本文将Cn中的Roper-Suffridge算子推广到任意复Banach空间中,并证明这种算子在任意复Banach空间中的某些区域上具有保持ε星形性,由此可以构造出任意复Banach空间,复Hilbert空间和Cn中的一些区域上的许多双全纯星形映照、双全纯凸映照、双全纯ε星形映照,同时,得到它们的增长定理等,将龚升与刘太顺,Roper与suffridge,Graham,Kohr等学者在Cn中的一些结果推广到任意复Banach空间或复Hilbert空间中．     

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.     

THE EXTENSION OPERATOR IN BANACH SPACES FOR LOCALLY BIHOLOMORPHIC MAPPINGS

In this article, the generalized Roper-Suffridge extension operator in Banach spaces for locally biholomorphic mappings is introduced. It is proved that this operator preserves the starlikeness on some domains in Banach spaces but does not preserves convexity for some cases. Moreover, the growth theorem, covering theorem, and the radius of starlikeness are discussed. Some results of Roper and Suffridge, Gong and Liu, Graham et al in C^n are extended to Hilbert spaces or Banach spaces.     

若干Banach空间及有界凸域上的凸映射

本文研究了一类Ｂａｎａｃｈ空间上凸映射的性质。找出了一类Ｂａｎａｃｈ空间上单位球上凸映射的特征，并利用这些结果研究了一类有界凸域上的所有凸映射．     

Quasihyperbolic geodesics in convex domains

We show that quasihyperbolic geodesics exist in convex domains in reflexive Banach spaces and that quasihyperbolic geodesies are quasiconvex in the norm metric in convex domains in all normed spaces.     

Biholomorphic Functions in Dual Banach Spaces

We present an infinite-dimensional version of Cartan-Carathéodory-Kaup-Wu theorem about the analyticity of the inverse of a given analytic mapping. It is valid for a class of domains in separable Banach dual spaces that includes all bounded convex domains.     

Fixed point theorems for a class of nonlinear mappings related to maximal monotone operators in Banach spaces

In this paper, the class of nonspreading mappings in Banach spaces is introduced. This class contains the recently introduced class of firmly nonexpansive type mappings in Banach spaces and the class of firmly nonexpansive mappings in Hilbert spaces. Among other things, we obtain a fixed point theorem for a single nonspreading mapping in Banach spaces. Using this result, we also obtain a common fixed point theorem for a commutative family of nonspreading mappings in Banach spaces. Received: 10 August 2007     

On integral formulas and their applications in functional calculus

Integral formulas from the theory of holomorphic functions of several variables are applied to explicit treatment of the analytic functional calculus for commuting operators in Banach spaces (commutative Banach algebras). Among others the integral formula of Henkin in smooth convex domains, is used. An application is also given to commuting normal operators in Hilbert spaces.     

Fredholm-valued holomorphic mappings on a Banach space

In this article we show that the pointwise existence of a regulariser for holomorphic Fredhom-valued mappings defined on pseudo-convex domains in Banach spaces with an unconditional basis implies the existence of a holomorphic regulariser.     

Uniform domain representations of ℓp ‐spaces

The category of Scott‐domains gives a computability theory for possibly uncountable topological spaces, via representations. In particular, every separable Banach‐space is representable over a separable domain. A large class of topological spaces, including all Banach‐spaces, is representable by domains, and in domain theory, there is a well‐understood notion of parametrizations over a domain. We explore the link with parameter‐dependent collections of spaces in e. g. functional analysis through a case study of ?^p ‐spaces. We show that a well‐known domain representation of ?^p as a metric space can be made uniform in the sense of parametrizations of domains. The uniform representations admit lifting of continuous functions and are effective in p. Dependent type constructions apply, and through the study of the sum and product spaces, we clarify the notions of uniformity and uniform computability. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Central Subspaces of Banach Spaces

In this paper we study Banach spaces that admit weighted Chebyshev centres for finite sets. Such spaces have been extensively studied recently by Veselý using the approach of finitely intersecting balls. Following his approach we exhibit large classes of Banach spaces that have this property. Certain stability results for spaces of vector valued continuous and Bochner integrable functions are also obtained.     

内插空间理论的应用

综述了线性算子内插法与内插空间理论在Banach空间几何学,微分算子,逼近理论,积分算子,Fourier分析等领域的一些应用。     

∑e1型Banach空间上C0半群稳定性的谱特征

讨论一类不可分解的∑_e¹型Banach空间上有界线性算子的谱的特殊性质;给出了∑_e¹型Banach空间上一致有界C₀半群稳定性的一个谱特征,并给出稳定性定理的一个应用.     

A linear perturbed Palais-Smale condition for lower semicontinuous functions on Banach spaces

This paper introduces a notion of linear perturbed Palais-Smale condition for real-valued functions on Banach spaces. In terms of strongly exposed points, it presents a characterization which guarantees linear perturbed Palais-Smale condition holds for lower semicontinuous functions with bounded effective domains defined on a Banach space with the Radon-Nikody＇m property; and gives an example showing that linear perturbed P-S condition is strictly weaker than the P-S condition.     

Geometrical properties of nonlinear maps and their application, Part I: The topological degree of quasiruled Fredholm maps on quasicylindrical domains

In this paper we introduce the notion of quasicylindrical domains in Banach spaces and develop a concept of a degree for quasiruled Fredholm mappings on quasicylindrical domains. Note that this quasicylindrical structure appears in a rather natural way, whenever “analytically” given nonlinear pseudodifferential operators are investigated in spaces of sufficiently smooth functions. Moreover, the class of quasiruled Fredholm mappings on quasicylindrical domains is sufficiently large, so that within this framework one can study a quite large class of nonlinear boundary value problems which are related to pseudodifferential operators.     

The S-Related Dynamic Convex Valuation in the Brownian Motion Setting

Let be an array of row-wise exchangeable random elements in a separable Banach space. Strong laws of large numbers are obtained for under certain moment conditions on the random variables and a condition relating to nonorthogonality. By using reverse martingale techniques, similar results are obtained for triangular arrays of random elements inseparable Banach spaces which are row-wise exchangeable     

Best proximity points for asymptotic cyclic contraction mappings

In this paper, we prove the existence and convergence of best proximity points for asymptotic cyclic contractions in metric spaces with the property UC, as well as for asymptotic proximal pointwise contractions in uniformly convex Banach spaces. Moreover, we consider a generalized cyclic contraction mapping and prove the existence of best proximity points for such a mapping in Banach spaces which do not necessarily satisfy the geometric property UC.     

Fixed points of semigroups of Lipschitzian mappings defined on nonconvex domains

Certain fixed point theorems are established for nonlinear semigroups of Lipschitzian mappings defined on nonconvex domains in Hilbert and Banach spaces. Some known results are thus generalized.