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     

Isometries between function algebras with finite codimensional range

Let A be a function algebra on a compact space X. A linear isometry T of A into A is said to be codimension n or finite codimensional if the range of T has codimension n in A. In this paper we prove that such isometries can be represented as weighted composition mappings on a cofinite subset, (∂A)₀, of the Shilov boundary for A, ∂A. We focus on those finite codimensional isometries for which (∂A)₀=∂A. All the above results, applied to the particular case of codimension 1 linear isometries on C(X), are used to improve the classification provided by Gutek et al. in J. Funct. Anal. 101, 97–119 (1991). Received: 3 June 1998 / Revised version: 22 March 1999     

Convexity properties of generalizations of the arithmetic-geometric mean

In the eighteenth century, Landen, Lagrange and Gauss studied a function of two positive real numbers that has become known as the arithmetic-geometric mean (AGM). In the nineteenth century, Borchardt generalized the AGM to a function of any 2ⁿ(n = 1,2,3,…) positive real numbers. In this paper, we generalize the AGM to a function of any even number of positive real numbers. If M(a, b) is the original AGM then M(a, b) is concave in the pair (a, b) of positive numbers and log M(e^α, e^β) is convex in the pair (α,β) of real numbers; all our generalizations of the AGM behave similarly. We generalize this analysis extensively.     

Reich’s problem concerning Halpern’s convergence

We discuss Halpern’s convergence for nonexpansive mappings in Hilbert spaces. We prove that one of the conditions in [R. Wittmann, Approximation of fixed points of nonexpansive mappings, Arch. Math. (Basel), 58 (1992), 486–491] is the weakest sufficient condition among the conditions known to us. We also improve a necessary condition, which is close to Wittmann’s. This is one step to solve the problem raised by Reich in 1974 and 1983. Received: 15 July 2008     

Some results for a finite family of uniformly L-Lipschitzian mappings in Banach spaces

The purpose of this paper is to prove a strong convergence theorem for a finite family of uniformly L-Lipschitzian mappings in Banach spaces. The results presented in the paper improve and extend some recent results in Chang [1], Cho et al. [2] Ofoedu [5], Schu [7] and Zeng [8, 9]. The present studies were supported by “the Natural Science Foundation of China (No. 10771141),” the Natural Science Foundation of Zhejiang Province (Y605191), the Natural Science Foundation of Heilongjiang Province (A0211), the Scientific Research Foundation from Zhejiang Province Education Committee (20051897).     

Recurrence and ergodicity of interacting particle systems

Many interacting particle systems with short range interactions are not ergodic, but converge weakly towards a mixture of their ergodic invariant measures. The question arises whether a.s.the process eventually stays close to one of these ergodic states, or if it changes between the attainable ergodic states infinitely often (“recurrence”). Under the assumption that there exists a convergence–determining class of distributions that is (strongly) preserved under the dynamics, we show that the system is in fact recurrent in the above sense. We apply our method to several interacting particle systems, obtaining new or improved recurrence results. In addition, we answer a question raised by Ed Perkins concerning the change of the locally predominant type in a model of mutually catalytic branching. Received: 22 January 1999 / Revised version: 24 May 1999     

The weak lower semicontinuity of the Kobayashi distance and its applications

In this paper we prove the lower semicontinuity with respect to the weak topology of the Kobayashi distance in a bounded, convex and open subset of a reflexive Banach space. We apply this result to the Denjoy-Wolff theorem for condensing mappings in the unit open ball in a strictly convex reflexive Banach space with the Kadec-Klee property. Received June 30, 1998 / in final form December 20, 1999 / Published online July 20, 2000     

Completely Reducible Infinite-Dimensional Skew Linear Groups

 Let V be a left vector space over a division ring D and the group of all D-automorphisms of V. A subgroup G of is completely reducible of V is completely reducible as D–G bimodule. Our aim in this brief note is to point out that in a sense the very useful notion of a local marker extends from V finite-dimensional to V infinite-dimensional. (A local marker of a subgroup G of is any finitely generated subgroup X of G such that row n-space has least composition length as D–X bimodule. A local marker of G controls to a considerable extent the local behaviour of G.) Our main result is the following. Let G be a completely reducible subgroup of and let W be any finite-dimensional D-subspace of V. Then G has a finitely generated subgroup X such that for every finitely generated subgroup Y of G containing X the D–Y submodule WY has a D–Y submodule M with and completely reducible. We also give some examples and state without proof some stronger conclusions valid for various special subgroup G. (Received 21 December 1998; in revised form 31 May 1999)     

On universal subsets of Banach spaces

We prove that to most of the known hypercyclic operators A on separable Banach spaces there exist compact (compact convex, compact connected) subsets K of E such that each compact (compact convex, compact connected) subset of E can be approximated with respect to Hausdorff's distance by for suitable . Received July 8, 1997, in final form October 17, 1997     

The similarity degree of an operator algebra, II

For every integer , there is a unital closed subalgebra with similarity degree equal precisely to d, in the sense of our previous paper. This means that for any unital homomorphism we have with independent of u, and the exponent d in this estimate cannot be improved. The proof that the degree is larger than crucially uses an upper bound for the norms of certain Gaussian random matrices due to Haagerup and Thorbj?rnsen. We also include several complements to our previous publications on the same subject. Received: 25 April, 1999; Accepted: 23 June, 1999     

Elementary operators on antiliminal C^*-algebras

We establish the equivalence of the following three properties of a -algebra A. (a) Every positive elementary operator on A is completely positive. (b) The norm and the cb-norm coincide for every elementary operator on A. (c) A is an extension of an antiliminal -algebra by an abelian one. Received: 15 July 1998 / in revised form: 22 September 1998     

Estimates for the norm of powers of Volterra's operator through its singular values

We prove that for every Volterra operator T the inequality holds, where b is an absolute constant and are its singular values. Received January 26, 2000 / Published online February 5, 2001     

What Do Composition Operators Know About Inner Functions?

 This paper gives several different ways in which operator norms characterize those composition operators that arise from holomorphic self-maps ϕ of the unit disc that are inner functions. The setting is the Hardy space H ² of the disc, and the key result is a characterization of inner functions in terms of the asymptotic behavior of the Nevanlinna counting function. The case offers an interesting surprise. (Received 25 June 1999; in revised form 29 September 1999)     

Strongly relatively nonexpansive sequences in Banach spaces and applications

We introduce the concept of a strongly relatively nonexpansive sequence in a Banach space and investigate its properties. Then we apply our results to the problem of approximating a common fixed point of a countable family of relatively nonexpansive mappings in a uniformly convex and uniformly smooth Banach space.      

Strong convergence of Browder’s type iterations for left amenable semigroups of Lipschitzian mappings in Banach spaces

We study an iterative scheme of Browder’s type for a semigroup of Lipschitzian mappings satisfying some uniform Lipschitzian condition, from a compact convex subset C of a smooth Banach space into C, with respect to a sequence of strongly asymptotic invariant means defined on an appropriate space of bounded real valued functions of the semigroup. The results presented in this paper generalize the corresponding results of Lau, Miyake and Takahashi [Nonlinear Anal. 67 (2007), 1211–1225].      

On Landweber iteration for nonlinear ill-posed problems in Hilbert scales

Summary. In this paper we derive convergence rates results for Landweber iteration in Hilbert scales in terms of the iteration index for exact data and in terms of the noise level for perturbed data. These results improve the one obtained recently for Landweber iteration for nonlinear ill-posed problems in Hilbert spaces. For numerical computations we have to approximate the nonlinear operator and the infinite-dimensional spaces by finite-dimensional ones. We also give a convergence analysis for this finite-dimensional approximation. The conditions needed to obtain the rates are illustrated for a nonlinear Hammerstein integral equation. Numerical results are presented confirming the theoretical ones. Received May 15, 1998 / Revised version received January 29, 1999 / Published online December 6, 1999