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.

q–differ-integral operator on p–valent functions associated with operator on Hilbert space

Making use of multivalent functions with negative coefficients of the type f (z)=z^(p)-~(∑)_(k=p+1)^(∞)a_(k)z^(k),which are analytic in the open unit disk and applying the q-derivative a q–differintegral operator is considered.Furthermore by using the familiar Riesz-Dunford integral,a linear operator on Hilbert space H is introduced.A new subclass of p-valent functions related to an operator on H is defined.Coefficient estimate,distortion bound and extreme points are obtained.The convolution-preserving property is also investigated.

AREA INTEGRAL FUNCTIONS AND H∞ FUNCTIONAL CALCULUS FOR SECTORIAL OPERATORS ON HILBERT SPACES

Area integral functions are introduced for sectorial operators on Hilbert spaces. We establish the equivalence relationship between the square and area integral functions. This immediately extends McIntosh/Yagi＇s results on H∞ functional calculus of sectorial operators on Hilbert spaces to the case when the square functions are replaced by the area integral functions.

Regularity of Continuous Linear Operators on Banach Function Spaces

In this paper we present some characterizations of Banach function spaces on which every continuous linear operator is regular.

Characteristic conditions-for strongly continuous groups of linear operators on a Hilbert space

IN 1932, Stone [1] gave the first result concerning C0 semigroups generated by an unbounded operator,which says that a linear operator is the infinitesimal generator of a C0 group of unitary operators on a Hilbert space if and only if it is skew-adjoint. This result has been applied extensively to linear partial differential equations （PDEs） with a law of conservation. Hille [2] then discovered the generation theorem of the Hille-Yosida type for a C0 group on a Banach space. There are also some conditions under which a C0 semigroup on a Banach space can be embedded in a C0 group. We refer the readers to ref. [3] for details of the results mentioned above.

Perturbations of Generalized Frames in a Hilbert Space

In the paper, we study the stability of Hilbert space generalized frames under perturbations. We get some results that are in spirit close to classical results for discrete frames, due to OLE Christensen.

On the Finite-dimensional Optimal Approximations to the Infinite dimensional Linear Operators

A method of approaching to the infinite dimensional linear operators by the finite dimensional operators is discussed. It is shown that,for every infinite dimensional operator A and every natural number n, there exists an n dimensional optimal approximation to A. The norm error is found and the necessary and sufficient condition for such n dimensional optimal approximations to be unique is obtained.

The Representation of Continuous Linear Operator by Integral

In this paper,we study the representation of linear operator on but abandom the Radon-Nikodym property and give a necessary and sufficient condition for representability of linear operator on by integral.

Perturbation Analysis of Moore–Penrose Quasi-linear Projection Generalized Inverse of Closed Linear Operators in Banach Spaces

In this paper, we investigate the perturbation problem for the Moore-Penrose bounded quasi-linear projection generalized inverses of a closed linear operaters in Banach space. By the method of the perturbation analysis of bounded quasi-linear operators, we obtain an explicit perturbation theorem and error estimates for the Moore-Penrose bounded quasi-linear generalized inverse of closed linear operator under the T-bounded perturbation, which not only extend some known results on the perturbation of the oblique projection generalized inverse of closed linear operators, but also extend some known results on the perturbation of the Moore-Penrose metric generalized inverse of bounded linear operators in Banach spaces.

A New Self Adaptive Inertial Subgradient Extragradient Iterative Algorithm for Solving Variational Inequalities of Pseudomonotone Mappings in Real Hilbert Spaces

The main purpose of this paper is to introduce and deal with a self adaptive inertial subgradient extragradient iterative algorithm with a new and interesting stepsize rule in real Hilbert spaces.Under some proper control conditions imposed on the coefficients and operators,we prove a new strong convergence result for solving variational inequalities with regard to pseudomonotone and Lipschitzian operators.Moreover,some numerical simulation results are given to show the rationality and validity of our algorithm.

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 x Y = {{x,y}lx E X,y C Y}. We may view M as a multi-valued linear operator from X to Y by taking M（x） = {yl（x,y} C 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.

ON PROBABILISTIC NORM OF A LINEAR OPERATORS AND SPACE OF OPERATORS

Since the PN space (E, F) which satisfies condition (PN-5) is just a Menger PN-space (E, F, min), the results with regard to probabilistic norms of linear operators on PN-spaces obtained by Xiao Jianzhong have bigger limitations. In this paper, problems respecting probabilistic norms of linear operators and spaces of operators are studied on more general Menger PN-spaces. The results presented improve and generalize the corresponding results by Xiao.

A Quantitative Version of the Bishop–Phelps Theorem for Operators in Hilbert Spaces

In this paper, with the help of spectral integral, we show a quantitative version of the Bishop-Phelps theorem for operators in complex Hilbert spaces. Precisely, let H be a complex Hilbert space and 0 〈 s 〈 1/2. Then for every bounded linear operator T ： H → H and x0 ∈ H with ｜｜T｜｜ = 1 = ｜｜xo｜｜ such that ｜｜Txo｜｜ 〉 1-6, there exist xε ∈ H and a bounded linear operator S ： H → H with ｜｜S｜｜ = 1 = ｜｜xε｜｜ such that ｜｜Sxε｜｜=1, ｜｜x-ε0｜｜≤√2ε＋4√2ε, ｜｜S-T｜｜≤√2ε.

Separability for Positive Operators on Tensor Product of Hilbert Spaces

The separability and the entanglement(that is,inseparability)of the composite quantum states play important roles in quantum information theory.Mathematically,a quantum state is a trace-class positive operator with trace one acting on a complex separable Hilbert space.In this paper,in more general frame,the notion of separability for quantum states is generalized to bounded positive operators acting on tensor product of Hilbert spaces.However,not like the quantum state case,there are different kinds of separability for positive operators with different operator topologies.Four types of such separability are discussed;several criteria such as the finite rank entanglement witness criterion,the positive elementary operator criterion and PPT criterion to detect the separability of the positive operators are established;some methods to construct separable positive operators by operator matrices are provided.These may also make us to understand the separability and entanglement of quantum states better,and may be applied to find new separable quantum states.

Inequalities and Separation for a Biharmonic Laplace-Beltrami Differential Operator in a Hilbert Space Associated with the Existence and Uniqueness Theorem

In this paper, we have studied the separation for the biharmonic Laplace-Beltrami differential operatorAu（x） = -△△u（x） ＋ V（x）u（x）,for all x ∈ R^n, in the Hilbert space H = L2（R^n,H1） with the operator potential V（x） ∈ C^1 （R^n, L （H1） ）, where L （H1 ） is the space of all bounded linear operators on the Hilbert space H1, while AAu is the biharmonic differential operator and△u=-∑i,j=1^n 1/√detg δ/δxi[√detgg-1（x）δu/δxj]is the Laplace-Beltrami differential operator in R^n. Here g（x） = （gij（x）） is the Riemannian matrix, while g^-1 （x） is the inverse of the matrix g（x）. Moreover, we have studied the existence and uniqueness Theorem for the solution of the non-homogeneous biharmonic Laplace-Beltrami differential equation Au = - △△u ＋ V（x） u （x） = f（x） in the Hilbert space H where f（x） ∈ H as an application of the separation approach.

A Gradient Iteration Method for Functional Linear Regression in Reproducing Kernel Hilbert Spaces

We consider a gradient iteration algorithm for prediction of functional linear regression under the framework of reproducing kernel Hilbert spaces.In the algorithm,we use an early stopping technique,instead of the classical Tikhonov regularization,to prevent the iteration from an overfitting function.Under mild conditions,we obtain upper bounds,essentially matching the known minimax lower bounds,for excess prediction risk.An almost sure convergence is also established for the proposed algorithm.

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）.

The Equivalence Relations between the Best Approximation and Linear Operators Approximation in a Besov Space

In this paper, we characterize Besov type space introduced by the best approximation, best approximating elements and a kind of linear operators.

CERTAIN SUBCLASSES OF ANALYTIC p－VALENT FUNCTIONS FOR OPERATORS ON HILBERT SPACE

In this paper,We mainly discuss some subclasses of analytic functions for operators onac, and determine a representation formula and sufficient condition for the class 4;(a.p,r,E,7) and prove a sufficient and necessary condition.coefficient estimates and distortiontheorem for the subclass O;o (a,p,Y,e,9) of the class ac;(a,p,y,c,V).Furthermore,the analogues also are true for operators on Hilbert space.

Composition Operators with Linear Fractional Symbols on Vector-Valued Bergman Spaces

Let φ and ψ be linear fractional self\|maps of the unit disk D and X a separable Hilbert space. In this paper we completely characterize the weak compactness of the product operators of a composition operation C φ with another one's adjoint C * ψ on the vector\|valued Bergman space B 1(X) for forms C φC * ψ and C * ψC φ.