New Estimates for the Rate of Convergence of the Method of Subspace Corrections

We discuss estimates for the rate of convergence of the method of successive subspace corrections in terms of condition number estimate for the method of parallel subspace corrections. We provide upper bounds and in a special case, a lower bound for preconditioners defined via the method of successive subspace corrections.     

THE CONTRACTION LINEAR METRIC PROJECTIONS IN Lp SPACES

In this paper ,we study the contraction lineartiy for metric projection in Lp spaces,A geometrical property of a subspace Y of Lp is given on which Pr is a contraction projection.     

A NOTE ON COMPLEX STRICTLY CONVEXITY AND COMPLEX SMOOTHNESS

In this paper, We show that X“ is complex strictly convex if and only if every quotient subspace of X is complex smooth, This extends a main theorem in [1].     

A Class Of Counterexamples Concerning an Open Problem

Kenneth R. Davidson raised ten open problems in the book Nest Algebras. One of theseopen problems isProblem 7 If K（交集）AlgL is weak^* dense in AlgL, where K is the set of all compact operators in B(H）,is L completely distributive? In this note, we prove that there is a reflexive subspace lattice L on some Hilbert space, which satisfies the following conditions: (a)F(AlgL) is dense in AlgL in the ultrastrong operator topology, where F(AlgL) is the set of all finite rank operators in AlgL; (b) L isn‘t a completely distributive lattice. The subspace lattices that satisfy the above conditions form a large class of lattices. As a special case of the result, it easy to see that the answer to Problem 7 is negative.     

A NOTE ON REARRANGEMENTS OF SERIES IN BANACH SPACES

We generalize Kadec‘s and Yan Zikun‘s resuhs in [1],[2] to prove; in every infinite dimensional real Banach space Y there exist an infinite dimensional subspace X and a series ∑xn such that ∑xn has property(N) in X. but(the domain of sums of this series) S(∑xn) is not a convex set.     

The finite-dimensional decomposition property in non-Archimedean Banach spaces

A non-Archimedean Banach space has the orthogonal finite-dimensional decomposition property（OFDDP） if it is the orthogonal direct sum of a sequence of finite-dimensional subspaces.This property has an influence in the non-Archimedean Grothendieck＇s approximation theory,where an open problem is the following： Let E be a non-Archimedean Banach space of countable type with the OFDDP and let D be a closed subspace of E.Does D have the OFDDP？ In this paper we give a negative answer to this question; we construct a Banach space of countable type with the OFDDP having a one-codimensional subspace without the OFDDP.Next we prove that,however,for certain classes of Banach spaces of countable type,the OFDDP is preserved by taking finite-codimensional subspaces.     

CHARACTERIZATION OF BEST APPROXIMATIONS IN METRIC LINEAR SPACES

Let (X,d) be a real metric linear space, with translation-invariant metric d and G a linear subspace of X. In this paper we use functionals in the Lipschitz dual of X to characterize those elements of G which are best approximations to elements of X. We also give simultaneous characterization of elements of best approximation and also consider elements of ε-approximation.     

The α-orthogonal complements of regular Dirichlet subspaces for one-dimensional Brownian motion

Roughly speaking a regular Dirichlet subspace of a Dirichlet form is a subspace which is also a regular Dirichlet form on the same state space. In particular, the domain of regular Dirichlet subspace is a closed subspace of the Hilbert space induced by the domain and α-inner product of original Dirichlet form. We investigate the orthogonal complement of regular Dirichlet subspace for one-dimensional Brownian motion in this paper. Our main results indicate that this orthogonal complement has a very close connection with theα-harmonic equation under Neumann type condition.     

τ-CHEBYSHEV AND τ-COCHEBYSHEV SUBPSACES OF BANACH SPACES

The concepts of quasi-Chebyshev and weakly-Chebyshev and σ-Chebyshev were defined [3 - 7], andas a counterpart to best approximation in normed linear spaces, best coapprozimation was introduced by Franchetti and Furi^[1]. In this research, we shall define τ-Chebyshev subspaces and τ-cochebyshev subspaces of a Banach space, in which the property τ is compact or weakly-compact, respectively. A set of necessary and sufficient theorems under which a subspace is τ-Chebyshev is defined.     

Global Existence in H^2 for a 1-d Non-monotone Fluid

In this paper, we prove the existence in H＋^2, an incomplete metric subspace of H^2×H^2×H^2, of global solutions to the system for a one-dimensional non-monotone fluid in bounded domainΩ=（0,1）. The results in this paper have improved those previously related results.     

The Representive of Metric Pro jection on the Finite Co dimension Subspace in Banach Space

In the paper we introduce the notions of the separation factor κ and give a representive of metric projection on an n-codimension subspace(or an affine set)under certain conditions in Banach space. Further, we obtain the distance formula from any point x to a finite n-codimension subspace. Results extend and improve the corresponding results in Hilbert space.     

ASYMPTOTICALLY ISOMETRIC COPIES OF In （1≤ p 〈 ∞） AND co IN BANACH SPACES

Let X be a Banach space. If there exists a quotient space of X which is asymptotically isometric to l^1, then X contains complemented asymptotically isometric copies of l^1. Every infinite dimensional closed subspace of l1. contains a complemented subspace of l1 which is asymptotically isometric to l1. Let X be a separable Banach space such that X^＊ contains asymptotically isometric copies of lp （1 〈 p 〈∞）. Then there exists a quotient space of X which is asymptotically isometric to lq （1/p ＋ 1/q=1）. Complemented asymptotically isometric copies of co in K（X, Y） and W（X, Y） are discussed. Let X be a Gelfand-Phillips space. If X contains asymptotically isometric copies of co, it has to contain complemented asymptotically isometric copies of co.     

Criteria for the single-valued metric generalized inverses of multi-valued linear operators in banach spaces

Let X, Y be Banach spaces and M be a linear subspace in X × Y = {{x, y}|x ∈ X, y ∈ Y }. We may view M as a multi-valued linear operator from X to Y by taking M (x) = {y|{x, y} ∈ M }. In this paper, we give several criteria for a single-valued operator from Y to X to be the metric generalized inverse of the multi-valued linear operator M . The principal tool in this paper is also the generalized orthogonal decomposition theorem in Banach spaces.     

AN ITERATED-SUBSPACE MINIMIZATION METHODS WITH SYMMETRIC RANK-ONE UPDATING

We consider an Iterated-Subspace Minimization(ISM) method for solving large-scale unconstrained minimization problems.At each major iteration of the method, a two-dimensional manifold, the iterated subspace, is constructed and an approximate minimizer of the objective function in this manifold then determined, and a symmetric rank-one updating is used to solve the inner minimization problem.     

CONVERGENCE ANALYSIS ON ITERATIVE METHODS FOR SEMIDEFINITE SYSTEMS

The convergence analysis on the general iterative methods for the symmetric and positive semidefinite problems is presented in this paper. First, formulated are refined necessary and sumcient conditions for the energy norm convergence for iterative methods. Some illustrative examples for the conditions are also provided. The sharp convergence rate identity for the Gauss-Seidel method for the semidefinite system is obtained relying only on the pure matrix manipulations which guides us to obtain the convergence rate identity for the general successive subspace correction methods. The convergence rate identity for the successive subspace correction methods is obtained under the new conditions that the local correction schemes possess the local energy norm convergence. A convergence rate estimate is then derived in terms of the exact subspace solvers and the parameters that appear in the conditions. The uniform convergence of multigrid method for a model problem is proved by the convergence rate identity. The work can be regradled as unified and simplified analysis on the convergence of iteration methods for semidefinite problems [8, 9].     

Maximal subspaces for solutions of the second order abstract cauchy problem

For a continuous, increasing function ω: R → R \{0} of finite exponential type, this paper introduces the set Z(A, ω) of all x in a Banach space X for which the second order abstract differential equation (2) has a mild solution such that [ω(t)]-1u(t,x) is uniformly continues on R , and show that Z(A, ω) is a maximal Banach subspace continuously embedded in X, where A ∈ B(X) is closed. Moreover, A|z(A,ω) generates an O(ω(t))strongly continuous cosine operator function family.     

INVARIANT SUBSPACES AND EIGEVALUES OF ELEMENTS IN C-ALGEBRAS

Let A be a C~*-algebra and x an element in A. the following invariant subspace problem is considered: Does there exist an irreducible representation π of A such that π(x) has a non-trivial invarint subspace? And a positive solution of the problem for finite separable matroid C~*-algebras is given. Also the eigenvalues Of elements in C~*-algebras is considered. Some versions of Fredholm Alternatives are given.     

LTV度量反馈控制问题的最优控制器

In this paper,we consider the measurement feedback control problem for discrete linear time-varying systems within the framework of nest algebra consisting of causal and bounded linear operators.Based on the inner-outer factorization of operators,we reduce the control problem to a distance from a certain operator to a special subspace of a nest algebra and show the existence of the optimal LTV controller in two different ways：one via the characteristic of the subspace in question directly,the other via the duality theory.The latter also gives a new formula for computing the optimal cost.     

RETRACTED ARTICLE: Some applications of BP-theorem in approximation theory

In this paper we apply Bishop-Phelps property to show that if X is a Banach space and G X is the maximal subspace so that G⊥ = {x* ∈ X*|x*(y) = 0; y∈ G} is an L-summand in X*, then L1(Ω,G) is contained in a maximal proximinal subspace of L1(Ω,X).     

The hyperspace of the regions below continuous maps with the fell topology

For a Tychonoff space X,we use ↓USC F(X) and ↓C F(X) to denote the families of the hypographs of all semi-continuous maps and of all continuous maps from X to I = [0,1] with the subspace topologies of the hyperspace Cld F(X × I) consisting of all non-empty closed sets in X × I endowed with the Fell topology.In this paper,we shall show that there exists a homeomorphism h:↓USC F(X) → Q = [1,1] ω such that h(↓CF(X))=c0 = {(xn)∈Q| lim n→∞ x n = 0} if and only if X is a locally compact separable metrizable space and the set of isolated points is not dense in X.