RELATIVE ESSENTIAL SPECTRA INVOLVING RELATIVE DEMICOMPACT UNBOUNDED LINEAR OPERATORS

In this article, we introduce the concept of demicompactness with respect to a closed densely defined linear operator, as a generalization of the class of demicompact operator introduced by Petryshyn in [24] and we establish some new results in Fredholm theory. Moreover, we apply the obtained results to discuss the incidence of some perturbation results on the behavior of relative essential spectra of unbounded linear operators acting on Banach spaces. We conclude by characterizations of the relative Schechter＇s and approximate essential spectrum.     

Continuous homogeneous selections of set-valued metric generalized inverses of linear operators in Banach spaces

In this paper,continuous homogeneous selections for the set-valued metric generalized inverses T of linear operators T in Banach spaces are investigated by means of the methods of geometry of Banach spaces.Necessary and sufficient conditions for bounded linear operators T to have continuous homogeneous selections for the set-valued metric generalized inverses T are given.The results are an answer to the problem posed by Nashed and Votruba.     

ON GRAPH MEASURES IN BANACH SPACES AND DESCRIPTION OF ESSENTIAL SPECTRA OF MULTIDIMENSIONAL TRANSPORT EQUATION

In this paper,we define new measures called respectively graph measure of noncompactness and graph measure of weak noncompactness.Moreover,we apply the obtained results to discuss the incidence of some perturbation results realized in [2] on the behavior of essential spectra of such closed densely defined linear operators on Banach spaces.These results are exploited to investigate the essential spectra of a multidimensional neutron transport operator on L1 spaces.     

Existence of multiple fixed points for nonlinear operators and applications

In this paper, by the fixed point index theory, the number of fixed points for sublinear and asymptotically linear operators via two coupled parallel sub-super solutions is studied. Under suitable conditions, the existence of at least nine or seven distinct fixed points for sublinear and asymptotically linear operators is proved. Finally, the theoretical results are applied to a nonlinear system of Hammerstein integral equations.     

Linear isometries of Hardy spaces

According to results established by DeLeeuw-Rudin-Wermer and by Forelli,all linear isometries of any Hardy space H~p(p≥1,p≠2)on the open unit discΔof C are represented by weighted composition operators defined by inner functions onΔ.After reviewing(and completing when p=∞)some of those results,the present report deals with a characterization of periodic and almost periodic semigroups of linear isometries of H~p.     

Linear Maps Preserving Invertibility or Related Spectral Properties

We survey some recent results on linear maps on operator algebras that preserve invertibility. We also consider related problems such as the problem of the characterization of linear maps preserving spectrum, various parts of spectrum, spectral radius, quasinilpotents, etc. We present some results on elementary operators and additive operators preserving invertibility or related properties. In particular, we give a negative answer to a problem posed by Gao and Hou on characterizing spectrumpreserving elementary operators. Several open problems are also mentioned.     

ON THE GENERALIZED RESOLVENT OF LINEAR PENCILS IN BANACH SPACES

Utilizing the stability characterizations of generalized inverses of linear operator, we investigate the existence of generalized resolvent of linear pencils in Banach spaces. Some practical criterions for the existence of generalized resolvents of the linear pencil λ→ T λ S are provided and an explicit expression of the generalized resolvent is also given. As applications, the characterization for the Moore-Penrose inverse of the linear pencil to be its generalized resolvent and the existence of the generalized resolvents of linear pencils of finite rank operators, Fredholm operators and semi-Fredholm operators are also considered. The results obtained in this paper extend and improve many results in this area.     

Perturbations of Moore–Penrose Metric Generalized Inverses of Linear Operators in Banach Spaces

In this paper,the perturbations of the Moore–Penrose metric generalized inverses of linear operators in Banach spaces are described.The Moore–Penrose metric generalized inverse is homogeneous and nonlinear in general,and the proofs of our results are different from linear generalized inverses.By using the quasi-additivity of Moore–Penrose metric generalized inverse and the theorem of generalized orthogonal decomposition,we show some error estimates of perturbations for the singlevalued Moore–Penrose metric generalized inverses of bounded linear operators.Furthermore,by means of the continuity of the metric projection operator and the quasi-additivity of Moore–Penrose metric generalized inverse,an expression for Moore–Penrose metric generalized inverse is given.     

Mild Solutions of a Class of Conformable Fractional Differential Equations with Nonlocal Conditions

This paper deals with the existence, uniqueness and continuous dependence of mild solutions for a class of conformable fractional differential equations with nonlocal initial conditions. The results are obtained by means of the classical fixed point theorems combined with the theory of cosine family of linear operators.     

关于线性正算子逼近的若干结果

In this paper we consider the degree of approximation of continuous function and intergrable function on the space with dimension m≥2 by the sequences of positive linear operators. Besides, we prove the Kolovkin type theorem for point-wise convergence of the sequences of positive linear operators.     

The representations of generalized inverses of lower triangular operators

In this note, the explicit representations of Moore-Penrose inverses of lower triangular operator matrices are established. As an application, the explicit representations of Bott-Duffin inverses of bounded linear operators on a Hilbert space with respect to a closed subspace are obtained.     

W_2~m空间中样条插值算子与线性泛函的最佳逼近

In this paper,the convergency of spline interpolation operators is obtained,these spline operators are determined by linear differential operators and constraint functionals.The errors of the interpolating spline with EHB fanctionals are estimated.The best approximation of linear functionals on W2^n spaces are investigated,which let to a useful computational method for the approximation solution of higher order linear differential equations with multipoint boundary value conditions.     

The closures of (U+K)-orbits of essentially normal triangular operator models

The (U + K)-orbit of a bounded linear operator T acting on a Hilbert space H is defined as (U + K)(T)={R-1 T R:R is invertible of the form unitary plus compact on H}.In this paper,we first characterize the closure of the (U + K)-orbit of an essentially normal triangular operator T satisfying H={ker(T-λI):λ∈ρ F (T)} and σ p (T*)=ф.After that,we establish certain essentially normal triangular operator models with the form of the direct sums of triangular operators,adjoint of triangular operators and normal operators,show that such operator models generate the same closed (U + K)-orbit if they have the same spectral picture,and describe the closures of the (U + K)-orbits of these operator models.These generalize some known results on the closures of (U + K)-orbits of essentially normal operators,and provide more positive cases to an open conjecture raised by Marcoux as Question 2 in his article "A survey of (U + K)-orbits".     

DEGREE OF APPROXIMATION ASSOCIATED WITH SOME ELLIPTIC OPERATORS AND ITS APPLICATIONS

The Jackson-type estimates by using some elliptic operators will be achieved. These results will be used to characterize the regularity of some elliptic operators by means of the approximation degree and the saturation class of the multivariate Bernstein operators as well.     

DIRECT AND CONVERSE RESULTS FOR LINEAR COMBINATIONS OF BASKAKOV-DURRMRYER OPERATORS

We consider the linear combinations of Baskakov-Durrmeyer operators and give the characterization interms of the classical modulous of smoothness for ∞-norm by the means of the pointwise simultanrous ap-proximation.An equivalence ralalion between the derivatives of these operators and smoothness or functions isalso presented.     

Property （ω） and topological uniform descent

We give the necessary and sufficient condition for a bounded linear operator with property （ω） by means of the induced spectrum of topological uniform descent, and investigate the permanence of property （ω） under some commuting perturbations by power finite rank operators. In addition, the theory is exemplified in the case of algebraically paranormal operators.     

POSITIVE SOLUTIONS TO A FIRST ORDER IMPULSIVE INTEGRAL BOUNDARY VALUE PROBLEM ON TIME SCALES

In this paper, we are mainly concerned with the existence and multiplicity of positive solutions to a frst order integral boundary value problem with impulsive efects on time scales. Under some conditions concerning the frst eigenvalues corresponding to the relevant linear operators, we obtain the main results by Krasnoselskii-Zabreiko fxed point theorem.     

Some Remarks on the Restriction Theorems for the Maximal Operators on Rd

The aim of this paper is to give a simple proof of the restriction theorem for the maximal operators on the d-dimensional Euclidean space Rd, whose theorem was proved by Carro-Rodriguez in 2012.Moreover, we shall give some remarks of the restriction theorem for the linear and the multilinear operators by Carro-Rodriguez and Rodriguez, too.     

Perturbation of the Moore–Penrose Metric Generalized Inverse in Reflexive Strictly Convex Banach Spaces

Let X,Y be reflexive strictly convex Banach spaces,let T,δT:X→Y be bounded linear operators with closed range R(T).Put T=T+δT.In this paper,by using the concept of quasiadditivity and the so called generalized Neumman lemma,we will give some error estimates of the bounds of |T~M|.By using a relation between the concepts of the reduced minimum module and the gap of two subspaces,some new existence characterization of the Moore-Penrose metric generalized inverse T~M of the perturbed operator T will be also given.