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QUASI-LOCAL CONJUGACY THEOREMS IN BANACH SPACES

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Chinese Annals of mathematics,Series B 2005年第4期26卷 551-558页

作者： ZHANG WEIRONG ma jipu Department of mathematics Nanjing University Nanjing 210093 China.

Let f ： U（x0） belong to E → F be a C^1 map and f＇（x0） be the Frechet derivative of f at x0. In local analysis of nonlinear functional analysis, implicit function theorem, inverse function theorem, local surjectivity theorem, local injectivity theorem, and the local conjugacy theorem are well known. Those theorems are established by using the properties： f＇（x0） is double splitting and R（f＇（x）） ∩ N（T0^＋） = {0} near x0. However, in infinite dimensional Banach spaces, f＇（x0） is not always double splitting （i.e., the generalized inverse of f（x0） does not always exist）, but its bounded outer inverse of f＇（x0） always exists. Only using the C^1 map f and the outer inverse To^# of f（x0）, the authors obtain two quasi-local conjugacy theorems, which imply the local conjugacy theorem if x0 is a locally fine point of f. Hence the quasi-local conjugacy theorems generalize the local conjugacy theorem in Banach spaces.

关键词： Frechet derivative Quasi-local conjugacy theorems Outer inverse Local conjugacy theorem

A common property of R(E,F) and B(R^n,R^m) and a new method for seeking a path to connect two operators

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Science China mathematics 2010年第10期53卷 2605-2620页

作者： ma ZhaoFeng ma jipu Zhejiang Inst Commun Dept Informat & Management Hangzhou 311112 Zhejiang Peoples R China Harbin Normal Univ Tseng Yuan Rong Funct Anal Res Ctr Harbin 150025 Peoples R China

Given two Banach spaces E,F,let B(E,F) be the set of all bounded linear operators from E into F,and R(E,F) the set of all operators in B(E,F) with finite *** is well-known that B(Rn) is a Banach space as well as an algebra,while B(Rn,Rm) for m = n,is a Banach space but not an algebra;meanwhile,it is clear that R(E,F) is neither a Banach space nor an ***,in this paper,it is proved that all of them have a common property in geometry and topology,i.e.,they are all a union of mutual disjoint path-connected and smooth submanifolds (or hypersurfaces).Let Σr be the set of all operators of finite rank r in B(E,F) (or B(Rn,Rm)).In fact,we have that 1) suppose Σr∈ B(Rn,Rm),and then Σr is a smooth and path-connected submanifold of B(Rn,Rm) and dimΣr = (n + m)r-r2,for each r ∈ [0,min{n,m});if m = n,the same conclusion for Σr and its dimension is valid for each r ∈ [0,min{n,m}];2) suppose Σr∈ B(E,F),and dimF = ∞,and then Σr is a smooth and path-connected submanifold of B(E,F) with the tangent space TAΣr = {B ∈ B(E,F) : BN(A)-R(A)} at each A ∈Σr for 0 r mation of matrices in B(Rn),and more adapted and simple than the elementary transformation *** addition to tensor analysis and application of Thom’s famous result for transversility,these will benefit the study of infinite geometry.

关键词： operator of finite rank smooth Banach submanifold matrix

SMOOTH AND PATH CONNECTED BANACH SUBmaNIFOLD Σ_r OF B(E,F) AND A DIMENSION FORMULA IN B(R^n,R^m)

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Analysis in Theory and Applications 2008年第4期24卷 395-400页

作者： jipu ma Harbin Normal University/Nanjing University China

Given two Banach spaces E, F, let B（E, F） be the set of all bounded linear operators from E into F, ∑r the set of all operators of finite rank r in B（E, F）, and ∑r^# the number of path connected components of ∑r. It is known that ∑r is a smooth Banach submanifold in B（E,F） with given expression of its tangent space at each A ∈ ∑r. In this paper, the equality ∑r^# = 1 is proved. Consequently, the following theorem is obtained： for any nonnegative integer r,∑r is a smooth and path connected Banach submanifold in B（E, F） with the tangent space TA∑r = {B E B（E,F）： BN（A） belong to R（A）} at each A ∈ ∑r if dim F = ∞. Note that the routine method can hardly be applied here. So in addition to the nice topological and geometric property of ∑r the method presented in this paper is also interesting. As an application of this result, it is proved that if E = R^n and F = R^m, then ∑r is a smooth and path connected submanifold of B（R^n,R^m） and its dimension is dim ∑r = （m ＋ n）r- r^2 for each r, 0≤r 〈 min{n,m}.

关键词： operator of finite rank smooth Banach submanifolcl path connectivity perturbation analysis of generalized inverse

TOPOLOGICAL AND GEOMETRIC PROPERTY OF maTRIX ALGEBRA

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Analysis in Theory and Applications 2007年第2期23卷 196-200页

作者： jipu ma Tseng Yuan-Yun Functional Research Center Harbin Normal University Harbin 150080 E R. China

Let B（R^n） be the set of all n x n real matrices, Sr the set of all matrices with rank r, 0 ≤ r ≤ n, and Sr^# the number of arcwise connected components of Sr. It is well-known that Sn =GL（R^n） is a Lie group an... 详细信息

关键词： matrix algebra acrwise connected component smooth hypersurface (submanifold) generalized inverse dimension of hypersurface

(1.2)INVERSES OF OPERATORS BETWEEN BANACH SPACES AND LOCAL CONJUGACY THEOREM

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Chinese Annals of mathematics,Series B 1999年第1期20卷 57-62页

作者： ma jipu(Department of mathematics,Nanjing University,Nanjing 210093,China) Nanjing Univ Dept Math Nanjing 210093 Peoples R China

Let E and F be Banach spaces and f non-linear C1 map from E into F. The main result isTheorem 2.2, in which a connection between local conjugacy problem of f at x0E and a localfine property of f'(x) at x0(see the Definition 1.1 in this paper) are obtained. This theoremincludes as special cases the two known theorems: the finite rank theorem and Berger's Theoremfor non-linear Fredholm operators. Moreover, the thcorem gives rise the further results for somenon-linear semi-Fredholm maps and for all non-linear semi-Wedholm maps when E and F areHilbert spaces. Thus Theorem 2.2 not only just unifies the above known theorems but alsoreally generalizes them.

关键词： Nonlinear semi-Fredholm maps Conjugacy problem Banach space

Nonlinear ergodic theorem for semitopological semigroups of non-Lipschitzian mappings in Banach spaces

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Chinese Science Bulletin 1997年第1期42卷 8-11页

作者： LI Gang and ma jipu1. Department of mathematics, Yangzhou Normal College, Yangzhou 225002, China 2. Department of mathematics, Nanjing University, Nanjing 210093, China Department of mathematics Yangzhou Normal College Yangzhou China Department of mathematics Nanjing University Nanjing China

LET G be a semitopological semigroup, i. e. G is a semigroup with a Hausdorff topology such that for each s∈G the mapping s|→s·t and s|→t·s from G to G are continuous. G is called right reversible if any two closed left ideals of G have non-void intersection. In this case, (G,≤) is a directed system when the binary relation '≤' on G is defined by a≤b if and only if Let E be the uniformly convex Banach space

关键词： nonlinear semigroup ergodic theorem non-Lipschitzian mapping.

The generalized solution of ill-posed boundary problem

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Science China mathematics 2006年第7期49卷 902-911页

作者： CAO Weiping & ma jipu Department of mathematics and Physics, Huaihai Institute of Technology, Lianyungang 222005, China Y. Y. Tseng Functional Analysis Research Centre, Harbin Normal University, Harbin 150080, China 1. Department of mathematics and Physics Huaihai Institute of Technology Lianyungang 222005 China 2. Y. Y. Tseng Functional Analysis Research Centre Harbin Normal University Harbin 150080 China

In this paper, we define a kind of new Sobolev spaces, the relative Sobolev spaces Wk,p0(Ω,∑). Then an elliptic partial differential equation of the second order with an ill-posed boundary is discussed. By utilizing the ideal of the generalized inverse of an operator, we introduce the generalized solution of the ill-posed boundary problem. Eventually, the connection between the generalized inverse and the generalized solution is studied. In this way, the non-instability of the minimal normal least square solution of the ill-posed boundary problem is avoided.

关键词： generalized inverse of operator, generalized solution, ill-posed boundary problem, elliptic partial differential equations of second order.

QUASI-LOCAL CONJUGACY THEOREMS IN BANACH SPACES

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Chinese Annals of mathematics 2005年第4期 64-71页

作者： ZHANG WEIRONG ma jipu Department of mathematics, Nanjing University, Nanjing 210093, China. Department of mathematics, Nanjing University, Nanjing 210093, China. Department of mathematics Nanjing University Nanjing 210093 China.

Let f : U(x0) (?) E→F be a. C1 map and f’(X0) be the Prechet derivative of /fat X0. In local analysis of nonlinear functional analysis, implicit function theorem, inverse function theorem, local surjectivity theorem, local injectivity theorem, and the local conjugacy theorem are well known. Those theorems are established by using the properties: f’(x0) is double splitting and R(f’(x))∩N(T0+) = {0} near X0. However, in infinite dimensional Banach spaces, f’(x0) is not always double splitting (i.e., the generalized inverse of f’(xo) does not always exist), but its bounded outer inverse of f’(x0) always exists. Only using the C1 map f and the outer inverse T0# of f’(x0), the authors obtain two quasi-local conjugacy theorems, which imply the local conjugacy theorem if X0 is a locally fine point of f. Hence the quasi-local conjugacy theorems generalize the local conjugacy theorem in Banach spaces.

关键词： Frediet derivative Quasi-local conjugacy theorems Outer inverse Local conjugacy theorem

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论广义横截性

数学物理学报（A辑） 2007年第2期27卷 351-358页

作者：方正马吉溥江南大学理学院无锡214122 南京大学数学系南京210093

设M和N是Cr(r≥1)Banach流形,P■N是N的子流形,f是从M到N的C1映射．该文引进映射f在x0∈f-1(P)点与P广义横截的概念,它是经典的横截概念的推广．接着讨论了广义横截性和广义正则点的关系,证明:映射f在x0点与P广义横截的充分必要条件为x... 详细信息

设M和N是Cr(r≥1)Banach流形,P■N是N的子流形,f是从M到N的C1映射．该文引进映射f在x0∈f-1(P)点与P广义横截的概念,它是经典的横截概念的推广．接着讨论了广义横截性和广义正则点的关系,证明:映射f在x0点与P广义横截的充分必要条件为x0是与f相关的某个映射g的广义正则点;当子流形P退化成单点集时,若映射f与P={p}广义横截,作者证明p是f的广义正则值;最后证明了广义横截点的全体D={x∈f-1(P):f(?)GxP}是开集．

关键词： Banach流形广义逆广义横截广义正则值开集