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2.
Jipu Ma 《中国科学A辑(英文版)》1997,40(12):1233-1238
A generalized dimension is further developed. Here subtraction and addition of two generalized dimensions are defined, so
that the operations: ∞ ± n = ∞, ∞ + ∞ = ∞, which used to play an inflexible role, are refined and moreover, ∞ - ∞, which used
to be meaningless, is done in sense. Then generalized index for semi-Fred-holm operators is developed to wholeB(H), i.e. all of bounded linear operators in Hilbert spaceH. Theorem 2.2 is proved with an example, which is in contradiction to a known proposition for semi-Fredholm operators in form,
practically a refined result of the known proposition. Then, it is proved thatB(H) is the union of countably many disjoint arewise connected sets over all the generalized dimensions ofB(H).
Project supported by the National Natural Science Foundation of China 相似文献
3.
A generalized preimage theorem in global analysis 总被引:7,自引:0,他引:7
Ma Jipu 《中国科学A辑(英文版)》2001,44(3):299-303
The concept of locally fine point and generalized regular value of a C1 map between Banach spaces were carried over C1 map between Banach manifolds. Hence the preimage theorem, a principle constructing Banach manifolds in global analysis, is
generalized. 相似文献
4.
Jipu Ma 《Frontiers of Mathematics in China》2008,3(2):305-316
Let E and F be Banach spaces, f: U ⊂ E → F be a map of C
r
(r ⩾ 1), x
0 ∈ U, and ft (x
0) denote the FréLechet differential of f at x
0. Suppose that f′(x
0) is double split, Rank(f′(x
0)) = ∞, dimN(f′(x
0)) > 0 and codimR(f′(x
0)) s> 0. The rank theorem in advanced calculus asks to answer what properties of f ensure that f(x) is conjugate to f′(x
0) near x
0. We have proved that the conclusion of the theorem is equivalent to one kind of singularities for bounded linear operators,
i.e., x
0 is a locally fine point for f′(x) or generalized regular point of f(x); so, a complete rank theorem in advanced calculus is established, i.e., a sufficient and necessary condition such that the
conclusion of the theorem to be held is given.
相似文献
5.
Given two Banach spaces E,F, let B(E,F) be the set of all bounded linear operators from E into F, Σ
r
the set of all operators of finite rank r in B(E,F), and Σ
r
# the number of path connected components of Σ
r
. It is known that Σ
r
is a smooth Banach submanifold in B(E,F) with given expression of its tangent space at each A ∈ Σ
r
. In this paper,the equality Σ
r
# = 1 is proved. Consequently, the following theorem is obtained: for any nonnegative integer r, Σ
r
is a smooth and path connected Banach submanifold in B(E,F) with the tangent space T
A
Σ
r
= {B ∈ B(E,F): BN(A) ⊂ R(A)} at each A ∈ Σ
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 Σ
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 Σ
r
is a smooth and path connected submanifold of B(ℝ
n
, ℝ
m
) and its dimension is dimΣ
r
= (m+n)r−r
2 for each r, 0 <- r < min {n,m}.
Supported by the National Science Foundation of China (Grant No.10671049 and 10771101). 相似文献
6.
Ma Jipu 《数学年刊B辑(英文版)》1982,3(5):595-598
Let A be a linear bounded operator in Hilbert space H with polar respresentation A =J(A^*A)^{1/2} where J^2=I, J^* = J. we use \pho_J(A) to denote the set of all complex \lambda, such that for any $\lambda \in \rho_J(A)$ there exist an bounded inverce R_J(A,\lambda) of (A—\lambda J)and \sigma_J(A)to complement of \rho_J(A).
Let S be a closed Cauchy domain, S\supset \sigma_J(A) and f (z) an analytic function on S. We define
$f(A)=\frac{1}{2\pi i}\oint\limits_{2s} {f(\zeta ){R_J}(A,\zeta )d\zeta }$,
the set of all such f(A)is denoted M.
If f(z) be analytic on S and symmetrical for real axis then f(A) is J-self adjoint. The set of all such f(A)is denoted M'. Let A\otimes B = AJB for A, B \in M(or M'). We have
Theorem, the ring of functions analytic on S (or analytic symmetrical for real axis on S) is a algebra homomorphism of M (or M'). The constant function 1 or z corresponds to operator J or A^* respectively.
Let $M_J={JB|B \in M}$ and $M'_J={JB|B \in M'}$ If the spectrum of (A^*A)^{1/2} is detached, we have
Theorem. M_J has common non-trivial reducing subspace and it is true for M_J. 相似文献
7.
Jipu Ma 《Numerical Functional Analysis & Optimization》2013,34(2):225-239
In this article, a kind of nonregular constraint and a principle for seeking critical point under the constraint are presented, where no Lagrange multiplier is involved. Let E, F be two Banach spaces, g: E → F a c 1 map defined on an open set U in E, and the constraint S = the preimage g ?1(y 0), y 0 ∈ F. A main deference between the nonregular constraint and regular constraint is that g′(x) at any x ∈ S is not surjective. Recently, the critical point theory under the nonregular constraint is a concerned focus in optimization theory. The principle also suits the case of regular constraint. Coordinately, the generalized regular constraint is introduced and the critical point principle on generalized regular constraint is established. Let f: U → ? be a nonlinear functional. While the Lagrange multiplier L in classical critical point principle is considered, its expression is given by using generalized inverse g′+(x) of g′(x) as follows: if x ∈ S is a critical point of f| S , then L = f′(x) ○ g′+(x) ∈ F*. Moreover, it is proved that if S is a regular constraint, then the Lagrange multiplier L is unique; otherwise, L is ill-posed. Hence, in case of the nonregular constraint, it is very difficult to solve Euler equations; however, it is often the case in optimization theory. So the principle here seems to be new and applicable. By the way, the following theorem is proved: if A ∈ B(E, F) is double split, then the set of all generalized inverses of A, GI(A) is smooth diffeomorphic to certain Banach space. This is a new and interesting result in generalized inverse analysis. 相似文献
8.
Local conjugacy theorem, rank theorems in advanced calculus and a generalized principle for constructing Banach manifolds 总被引:6,自引:0,他引:6
MA Jipu 《中国科学A辑(英文版)》2000,43(12):1233-1237
Applications of locally fine property for operators are further developed. LetE andF be Banach spaces andF:U(x
0)⊂E→F be C1 nonlinear map, whereU (x
0) is an open set containing pointx
0∈E. With the locally fine property for Frechet derivativesf′(x) and generalized rank theorem forf′(x), a local conjugacy theorem, i. e. a characteristic condition forf being conjugate tof′(x
0) near x0,is proved. This theorem gives a complete answer to the local conjugacy problem. Consequently, several rank theorems in advanced
calculus are established, including a theorem for C1 Fredholm map which has been so far unknown. Also with this property the concept of regular value is extended, which gives
rise to a generalized principle for constructing Banach submanifolds. 相似文献
9.
The main purpose of this paper is to study the continuity of several kinds of generalized inverses of elements in a Banach algebra with identity. We first obtain a sufficient and necessary condition for the lower semi-continuity of reflexive generalized inverses as set-valued mappings. Based on this result, we characterize the continuity of the Moore-Penrose inverse in a C∗-algebra and therefore, derive some new and well-known criteria in operator theory. 相似文献
10.
In this paper we study the existence and multiplicity of solutions of the following operator equation in Banach space E:
u=λAu,0<λ<+∞,u∈P?{θ}, 相似文献