A NEW PERTURBATION THEOREM FOR MOORE-PENROSE METRIC GENERALIZED INVERSE OF BOUNDED LINEAR OPERATORS IN BANACH SPACES

In this paper, we investigate a new perturbation theorem for the Moore-Penrose metric generalized inverses of a bounded linear operator in Banach space. The main tool in this paper is ＂the generalized Neumann lemma＂ which is quite different from the method in [12] where ＂the generalized Banach lemma＂ was used. By the method of the perturba- tion analysis of bounded linear operators, we obtain an explicit perturbation theorem and three inequalities about error estimates for the Moore-Penrose metric generalized inverse of bounded linear operator under the generalized Neumann lemma and the concept of stable perturbations in Banach spaces.

Some Remarks on Operators and Their Conjugates

We first characterize the range spaces F for which {T＇：T∈K（E,F）}=K（F＇,E＇） holds for all Banach spaces E. Then we present some sufficient conditions for the equality T＇=T＇. Some related results are also included.

Several Norms of Operators on Banach Lattices

We present some positive results involving the several norms of operators on Banach lattices, which shows several relations of these operator norms.

Asymptotically Isometric Copy of l~β(0<β<1) in Spacesof Bounded Linear Operators

Assume X is a normed space,every x ＊ ∈ S（X＊） can reach its norm at some point in B（X）,and Y is a β-normed space.If there is a quotient space of Y which is asymptotically isometric to l β,then L（X,Y） contains an asymptotically isometric copy of l β.Some sufficient conditions are given under which L（X,Y） fails to have the fixed point property for nonexpansive mappings on closed bounded β-convex subsets of L（X,Y）.

Generalized regular points of a C^1 map between Banach spaces

Let f be a C^1 map between two Banach spaces E and F. It has been proved that the concept of generalized regular points of f, which is a generalization of the notion of regular points of f, has some crucial applications in nonlinearity and global analysis. We characterize the generalized regular points of f using the three integer-valued （or infinite） indices M（x0）, Mc（x0） and Mr（x0） at x0 ∈ E generated by f and by analyzing generalized inverses of bounded linear operators on Banach spaces, that is, iff ＇（x0） has a generalized inverse in the Banach space B（E, F） of all bounded linear operators on E into F and at least one of the indices M（x0）, Mc（x0） and Mr（x0） is finite, then xo is a generalized regular point off if and only if the multi-index （M（x）, Me（x）, Mr（x）） is continuous at X0.

SOME RESULTS RELATED TO THE SEPARABLE QUOTIENT PROBLEM

The problem whether every infinite dimensional Banach space has an infinite dimensional separable quotient space has remained unsolved for a long time. In this paper we prove: the Banach space X has an infinite dimensional separable quotient if and only if X has an infinite dimensional separable quasicomplemented subspace, also if and only if there exists a Banach space Y and a bounded linear operator T is an element of B(Y,X such that the range of T is nonclosed and dense in X. Besides, the other relevant questions for such spaces e.g. the question on operator ideals that on H.I.(hereditarily indecomposable) spaces, that on invariant subspaces of operators, etc, are also discussed.

THE METRIC GENERALIZED INVERSE AND ITS SINGLE-VALUE SELECTION IN THE PRICING OF CONTINGENT CLAIMS IN AN INCOMPLETE FINANCIAL MARKET

This article continues to study the research suggestions in depth made by M.Z.Nashed and G.F.Votruba in the journal"Bull.Amer.Math.Soc."in 1974.Concerned with the pricing of non-reachable"contingent claims"in an incomplete financial market,when constructing a specific bounded linear operator A:l_(1)^(n)→l_(2) from a non-reflexive Banach space l_(1)^(n) to a Hilbert space l_(2),the problem of non-reachable"contingent claims"pricing is reduced to researching the(single-valued)selection of the(set-valued)metric generalized inverse A■ of the operator A.In this paper,by using the Banach space structure theory and the generalized inverse method of operators,we obtain a bounded linear single-valued selection A^(σ)=A+of A■.

On Absolute N?rlund Spaces and Matrix Operators

In a more recent paper, the second author has introduced a space ｜Cα｜k as the set of all series by absolute summable using Cesaro matrix of order α 〉 -1. In the present paper we extend it to the absolute NSrlund space ｜Np^θ｜k taking Norlund matrix in place of Cesaro matrix, and also examine some topological structures, α-β-γ-duals and the Schauder base of this space. Further we characterize certain matrix operators on that space and determine their operator norms, and so extend some well-known results.

Some New Results Related to Compact Matrix Operators in the Class((l_p)T,l_∞)

The main goal of this paper is to establish necessary and sufficient conditions for a matrix A ∈（（lp）T,l∞）, where T is an arbitrary triangle, 1 ≤ p ≤∞, to be a compact operator. In the past,only sufficient conditions were established in almost all of those cases, by using the Hausdorff measure of noncompactness. We improve those results by applying another method for the characterizations of compact linear operators between BK spaces.