Iterative approximation to common fixed points of a countable family of nonexpansive mappings

We proved several strong convergence results by using the conception of a uniformly asymptotically regular sequence {T _n } of nonexpansive mappings in a reflexive Banach space which admits a weakly continuous duality mapping J _?(l ^p (1?p?t)?=?t ^p?1. The results presented develop and complement the corresponding ones by Song, Y. and Chen, R., 2007 [Iterative approximation to common fixed points of nonexpansive mapping sequences in reflexive Banach spaces. Nonlinear Analysis, 66, 591–603], Song, Y., Chen, R. and Zhou, H., 2007 [Viscosity approximation methods for nonexpansive mapping sequences in Banach spaces. Nonlinear Analysis, 66, 1016–1024] and O'Hara, J.G., Pillay, P. and Xu, H.K., 2006 [Iterative approaches to convex feasibility problem in Banach Space. Nonlinear Analysis, 64, 2022–2042], O'Hara, J.G., Pillay, P. and Xu, H.K., 2003 [Iterative approaches to fineding nearest common fixed point of nonexpansive mappings in Hilbert spaces. Nonlinear Analysis, 54, 1417–1426] and Jung, J.S., 2005 [Iterative approaches to common fixed points of nonexpansive mappings in Banach spaces. Journal of Mathematical Analysis and Applications, 302, 509–520] and many other existing literatures.     

Banach spaces without local unconditional structure

For a large class of Banach spaces, a general construction of subspaces without local unconditional structure is presented. As an application it is shown that every Banach space of finite cotype contains eitherl ₂ or a subspace without unconditional basis, which admits a Schauder basis. Some other interesting applications and corollaries follow. The contribution of this author is a part of his Ph.D. Thesis written at the University of Alberta under the supervision of the second author. An erratum to this article is available at .     

赋序列范数的矢值Banach序列空间ss(E)的凸性

本文对赋序列范数的矢值Banach序列空间ss(E)的一些凸性进行了讨论,得到的主要结果如下: 1.ss(E)是局部一致凸的当且仅当ss和E是局部一致凸的; 2.ss(E)是强凸的当且仅当ss和E是强凸的; 3.设ss和ss*具有AK性质,则ss(E)是非常凸的当且仅当ss和E是非常凸的.     

ON THE SPECTRAL RADIUS OF IRREDUCIBLE AND WEAKLY IRREDUCIBLE OPERATORS IN BANACH LATPICES

Abstract

If T is an operator on a Banach lattice E we call T weakly irreducible if E contains no non-trivial T-invariant bands. We prove that if E is order complete and if the weakly irreducible operator T > 0 is in (E′_oo ? E)^⊥⊥ then T has positive spectral radéus. Prom this follows that Jentesch's theorem holds in arbitrary Banach function spaces.

If [Ttilde] denotes the restriction of T′ to E′_oo, 0 ? T an order continuous operator, then T is weakly irreducible if and only if [Ttilde]: E′_oo→E′_oo is weakly irreducible.

Finally we show that the majorizing, irreducible operator T ≥ 0, has positive spectral radius if either Tⁿ is weakly compact or E has property (P) or T is strongly majorizing.     

A class of Banach sequence spaces analogous to the space of Popov

Hagler and the first named author introduced a class of hereditarily l ₁ Banach spaces which do not possess the Schur property. Then the first author extended these spaces to a class of hereditarily l _p Banach spaces for 1 ⩽ p < ∞. Here we use these spaces to introduce a new class of hereditarily l _p (c ₀) Banach spaces analogous of the space of Popov. In particular, for p = 1 the spaces are further examples of hereditarily l ₁ Banach spaces failing the Schur property.     

Extension of isometries between the unit spheres of complex l(Γ)(p>1)spaces

In this paper, we study the extension of isometries between the unit spheres of complex Banach spaces l^p(Γ) and l^p(Δ)(p > 1). We first derive the representation of isometries between the unit spheres of complex Banach spaces l^p(Γ) and l^p(Δ). Then we arrive at a conclusion that any surjective isometry between the unit spheres of complex Banach spaces l^p(Γ)and l^p(Δ) can be extended to be a linear isometry on the whole space.     

Iterative construction of fixed points of strictly pseudocontractive mappings

Let (E, ‖ ? ‖) be a smooth Banach space over the real field and A a nonempty closed bounded convex subset of E. Suppose T : A → A is a uniformly continuous strictly pseudocontractive selfmapping of A. Then, if [math001]satisfies [math001]the iteration process [math001] and [math001] converges strongly to the unique fixed point x of T. This is an improvement of a result of C.E. Chidume who established strong convergence of (x ⁿ to x in case E is L ^p or l ^p with [math001] making essential use of the inepuality [math001] which is kown to hold in these spaces for all x and y     

Locally uniformly rotund points in convex-transitive Banach spaces

Almost transitive superreflexive Banach spaces have been considered in [C. Finet, Uniform convexity properties of norms on superreflexive Banach spaces, Israel J. Math. 53 (1986) 81–92], where it is shown that they are uniformly convex and uniformly smooth. We characterize such spaces as those convex transitive Banach spaces satisfying conditions much weaker than that of uniform convexity (for example, that of having a weakly locally uniformly rotund point). We note that, in general, the property of convex transitivity for a Banach space is weaker than that of almost transitivity.     

Boundary conditions for linear manifolds. III. Infinite dimensional adjoints via solution spaces

A formula is given for the orthogonal complement of any vector subspace of l₂. Countably infinite adjoint subspaces in a Banach space are characterized via solution spaces. In particular, infinite dimensional self-adjoint subspaces in a reflexive Banach space are characterized via solution spaces, generalizing a result in Dunford and Schwartz [“Linear Operators, II,” Interscience, New York, 1963]. Applications are made to closed linear manifolds in l₂ ⊕ l₂ as well as infinite dimensional, generalized ordinary differential subspaces in a Hilbert space with the boundary conditions imposed on real sequences. The results are also expressed via solution spaces.     

Locally uniformly rotund renorming and fragmentability

In this paper we characterize those Banach spaces E that admita locally uniformly rotund renorming by means of a linear topologicalcondition that is a particular case of the notion, introducedby J. E. Jayne, I. Namioka and C. A. Rogers, called countablecover by sets of small local diameter. This allows us to developa Decomposition Method and a Transfer Technique to obtain locallyuniformly rotund renormings. 1991 Mathematics Subject Classification:46B20.     

A SUPPLEMENT TO MY PAPER ON REAL AND COMPLEX OPERATOR IDEALS

Abstract

We show that the main problem left open in [2] can be solved using the Banach spaces Z^α recently constructed by Kalton [1]. This gives an example of a complex operator ideal that has no real analogue. It thus shows the richer structure of complex operator ideals compared with the real ones.     

Some banach space properties of E φ [0, 1]

We prove several Banach spaces properties of E ^φ [0, 1], many of them concerning to the Δ₂ condition and the presence of c ₀ and l ₁ in those Banach spaces.     

A Characterization of l + 1(Γ)

In this article we give necessary and sufficient conditions in order a closed cone of a Banach space E to be isomorphic to the positive cone l ₁ ⁺(Γ) of an l ₁(Γ) space. This problem has applications in the theory of Banach spaces as in the characterization of reflexive spaces and also on the problem of the embeddability of 1in Banach spaces.     

On the projection and macphail constants ofl n p spaces

We prove that the projection and Macphail constants ofl _n ^p (1≦p≦2) are asymptotically equivalent ton ^1/2 andn ^−1/2 respectively. We also obtain some relations linking certain parameters of general finite dimensional real Banach spaces. This note is a part of the author’s Ph.D. Thesis prepared at the Hebrew University of Jerusalem, under the supervision of Prof. J. Lindenstrauss, to whom the author wishes to express his thanks and appreciation.     

On banach spaces which containl 1(τ) and types of measures on compact spaces

Two closely related results are presented, one of them concerned with the connection between topological and measure-theoretic properties of compact spaces, the other being a non-separable analogue of a result of Peŀczyński's about Banach spaces containingL ¹. Let τ be a regular cardinal satisfying the hypothesis that κ^ω<τ whenever κ<τ. The following are proved: 1) A compact spaceT carries a Radon measure which is homogeneous of type τ, if and only if there exists a continuous surjection ofT onto [0, 1]^τ. 2) A Banach spaceX has a subspace isomorphic tol ¹(τ) if and only ifX ^∗ has a subspace isomorphic toL ¹({0, 1}^τ). An example is given to show that a more recent result of Rosenthal's about Banach spaces containingl ¹ does not have an obvious transfinite analogue. A second example (answering a question of Rosenthal's) shows that there is a Banach spaceX which contains no copy ofl ¹ (ω₁), while the unit ball ofX ^∗ is not weakly^∗ sequentially compact.     

Approximative compactness and continuity of metric projector in Banach spaces and applications

First we prove that the approximative compactness of a nonempty set C in a normed linear space can be reformulated equivalently in another way.It is known that if C is a semi-Chebyshev closed and approximately compact set in a Banach space X,then the metric projectorπC from X onto C is continuous.Under the assumption that X is midpoint locally uniformly rotund,we prove that the approximative compactness of C is also necessary for the continuity of the projectorπC by the method of geometry of Banach spaces.Using this general result we find some necessary and sufficient conditions for T to have a continuous Moore-Penrose metric generalized inverse T~ ,where T is a bounded linear operator from an approximative compact and a rotund Banach space X into a midpoint locally uniformly rotund Banach space Y.     

Directions of uniform rotundity in direct sums of normed spaces

Let {(X_i,|| · || _i)}_{i ? I},\{(X_i,\left \| {\cdot } \right \| _{i})\}_{i\in I}, be an arbitrary family of normed spaces and let (E,|| · || _E)(E,\left \| {\cdot } \right \| _{E}) be a monotonic normed space of real functions on the set I that is an ideal in \Bbb R^I{\Bbb R}^I. We prove a sufficient condition for the direct sum space E(X_i) to be uniformly rotund in a direction. We show that this condition is also necessary for E=l_￥E=\ell _{\infty }, and it is not necessary for E=l₁E=\ell _1. When E is either uniformly rotund in every direction and has compact order intervals, or weakly uniformly rotund respect to its evaluation functionals, we reestablish as a corollary the result that reads: E(X_i)E(X_i) is uniformly rotund in every direction if and only if so are all the X_i.     

Spin structures on instanton moduli spaces

We determine the necessary and sufficient conditions for the based SU(l)-instanton moduli spaces over CP² and the unbased SU(2)-instanton moduli spaces over S⁴ having spin structures.     

Espaces de Banach stables

We define the notion of “stable Banach space” by a simple condition on the norm. We prove that ifE is a stable Banach space, then every subspace ofL ^p(E) (1≦p<∞) is stable. Our main result asserts that every infinite dimensional stable Banach space containsl ^p, for somep, 1≦p<∞. This is a generalization of a theorem due to D. Aldous: every infinite dimensional subspace ofL ¹ containsl ^p, for somep in the interval [1, 2].     

Espaces de Banach representables

We study the Banach spaces which are isomorphic to a subspace ofl ^∞(N) which is analytic inR ^N. We prove structure theorems which show that some pathological situations cannot take place in this class. We show that a non-metrizable separable compact of Rosenthal has a continuous image which is not a compact of Rosenthal.