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.

On n-K Width of Certain Function Classes Defined by Linear Differential Operators in L_2 Space

Let M(u) be an N function, A=D r+∑r-1k=0a k(x)D k a linear differential operator and W M(A) the Sobolev Orlicz class defined by M(u) and A. In this paper we give the asymptotic estimates of the n K width d n(W M(A),L 2[0,1]) .

On n-K Width of Certain Function Classes Defined by Linear Differential Operators in L2 Space

Let M(u) be an N function, A=D r+∑r-1k=0a k(x)D k a linear differential operator and W M(A) the Sobolev Orlicz class defined by M(u) and A. In this paper we give the asymptotic estimates of the n K width d n(W M(A),L 2[0,1]) .

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.

Linear Operators That Strongly Preserve Nilpotent Matrices over Antinegative Semirings

Let S be an antinegative commutative semiring without zero divisors and Mn（S） be the semiring of all n × n matrices over S. For a linear operator L on Mn（S）, we say that L strongly preserves nilpotent matrices in Mn（S） if for any A ∈ Mn（S）, A is nilpotent if and only if L（A） is nilpotent. In this paper, the linear operators that strongly preserve nilpotent matrices over S are characterized.

On n-K Width of Certain Function Classes Defined by Linear Operators in L_2 Space

Let M(u) be an N-function, L_r(f, x) and K_r(f, x) are Bak operator and Kantorovich operator, W_M(L_r(f)) and W_M(K_r(f)) are the Sobolev-Orlicz classes defined by L_r(f, x), K_r(f, x) and M(u). In this paper we give the asymptotic estimates of the n-K widths d_n(W_M(L_r(f)), L_2[0, 1]) and d_n(W_M(K_r(f)), L_2[0, 1]).

THE EXTENSION THEOREMS OF CONE LINEAR OPERATORS

A kind of cone separation theorems is established, by which the extension theorems for cone linear continuous operators are developed. As an application, the extension theorem for positive linear continuous operators is given.

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.

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.

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.

Linear Operators Preserving Symplectic Group over a Field Consisting of at Least Four Elements

Suppose F is a field consisting of at least four elements. Let Mn（F） and SP2n（F） be the linear space of all n × n matrices and the group of all 2n × 2n symplectic matrices over F, respectively. A linear operator L ： M2n（F） → M2n（F） is said to preserve the symplectic group if L（SP2n（F）） = SP2n（F）. It is shown that L is an invertible preserver of the symplectic group if and only if L takes the form （i） L（X） = QPXP^-1 for any X ∈ M2n（F） or （ii） L（X） = QPX^TP^-1 for any X ∈M2n（F）, where Q ∈ SP2n（F） and P is a generalized symplectic matrix. This generalizes the result derived by Pierce in Canad J. Math., 3（1975）, 715-724.

ON CONSERVATIVE APPROXIMATION BY LINEAR POLYNOMIAL OPERATORS AN EXTENSION OF THE BERNSTEIN'S OPERATOR

In this work we slwly linear polynomial operators preserving some consecutive i-convexities and leaving in-verant the polynomtals up to a certain degree. First we study the existence of an incompatibility between the conservation of cenain i-cotivexities and the invariance of a space of polynomials. Interpolation properties are obtained and a theorem by Berens and DcVore about the Bernstein's operator ts extended. Finally, from these results a genera'ized Bernstein's operator is obtained.

ESTIMATE ON THE BLOCH CONSTANT FOR CERTAIN HARMONIC MAPPINGS UNDER A DIFFERENTIAL OPERATOR

In this paper,we first obtain the precise values of the univalent radius and the Bloch constant for harmonic mappings of the formL(f)=zfz-zfz,where f represents normalized harmonic mappings with bounded dilation.Then,using these results,we present better estimations for the Bloch constants of certain harmonic mappings L(f),where f is a K-quasiregular harmonic or open harmonic.Finally,we establish three versions of BlochLandau type theorem for biharmonic mappings of the form L(f).These results are sharp in some given cases and improve the related results of earlier authors.

Spherical Linear Diophantine Fuzzy Sets with Modeling Uncertainties in MCDM

The existing concepts of picture fuzzy sets(PFS),spherical fuzzy sets(SFSs),T-spherical fuzzy sets(T-SFSs)and neutrosophic sets(NSs)have numerous applications in decision-making problems,but they have various strict limitations for their satisfaction,dissatisfaction,abstain or refusal grades.To relax these strict constraints,we introduce the concept of spherical linearDiophantine fuzzy sets(SLDFSs)with the inclusion of reference or control parameters.A SLDFSwith parameterizations process is very helpful formodeling uncertainties in themulti-criteria decisionmaking(MCDM)process.SLDFSs can classify a physical systemwith the help of reference parameters.We discuss various real-life applications of SLDFSs towards digital image processing,network systems,vote casting,electrical engineering,medication,and selection of optimal choice.We show some drawbacks of operations of picture fuzzy sets and their corresponding aggregation operators.Some new operations on picture fuzzy sets are also introduced.Some fundamental operations on SLDFSs and different types of score functions of spherical linear Diophantine fuzzy numbers(SLDFNs)are proposed.New aggregation operators named spherical linear Diophantine fuzzy weighted geometric aggregation(SLDFWGA)and spherical linear Diophantine fuzzy weighted average aggregation(SLDFWAA)operators are developed for a robust MCDM approach.An application of the proposed methodology with SLDF information is illustrated.The comparison analysis of the final ranking is also given to demonstrate the validity,feasibility,and efficiency of the proposed MCDM approach.

On the Stable Method Computing Values of Unbounded Operators

Unbounded operators can transform arbitrarily small vectors into arbitrarily large vectors—a phenomenon known as instability. Stabilization methods strive to approximate a value of an unbounded operator by applying a family of bounded operators to rough approximate data that do not necessarily lie within the domain of unbounded operator. In this paper we shall be concerned with the stable method of computing values of unbounded operators having perturbations and the stability is established for this method.

NEW SUBCLASS OF ANALYTIC FUNCTIONS IN CONICAL DOMAIN ASSOCIATED WITH A LINEAR OPERATOR

The main object of the present paper is to investigate a number of useful properties such as inclusion relations, distortion bounds, coefficient estimates, subordination results, the Fekete-Szego problem and some other for a new subclass of analytic functions, which are defined here by means of linear operator. Relevant connections of the results presented here with those obtained in earlier works are also pointed out.

PROBABILISTIC AND AVERAGE LINEAR WIDTHS OF SOBOLEV SPACE WITH GAUSSIAN MEASURE IN SPACE S_Q(T)(1≤Q≤∞)

Probabilistic linear （N, δ）-widths and p-average linear N-widths of Sobolev space W2^r（T）, equipped with a Gaussian probability measure #, are studied in the metric of Sq （T） （1 ≤ Q ≤∞）, and determined the asymptotic equalities：λN,δ（W2^r（T）,μ,Sq（T））={（N^-1）^r＋p/2-1/q√1＋1/N·ln1/δ, 1≤q≤2, （N^-1）^r＋p/2-1/q（1＋N^-1/q√ln1/δ）,2〈q〈∞, （N^-1）^r＋p/2√lnN/δ, q=∞,and λN^（a）（W2^r（T）,μ,Sq（T））p={（N^-1）^r＋p/2-1/q, 1≤q〈∞, （N^-1）^r＋p/2-1/q√lnN, q=∞,where 0 〈 p 〈 ∞, δ∈ （0, 1/2], ρ 〉 1, and Sq（T） is a subspace of L1（T）, in which the Fourier series is absolutely convergent in lq sense.

Control synthesis of linear distributed parameter switched systems

The control synthesis for switched systems is extended to distributed parameter switched systems in Hilbert space. Based on semigroup and operator theory, by means of multiple Lyapunov method incorporated average dwell time approach, sufficient con- ditions are derived in terms of linear operator inequalities frame- work for distributed parameter switched systems. Being applied to one dimensional heat propagation switched systems, these lin- ear operator inequalities are reduced to linear matrix inequalities subsequently. In particular, the state feedback gain matrices and the switching law are designed, and the state decay estimate is explicitly given whose decay coefficient completely depends on the system＇s parameter and the boundary condition. Finally, two numerical examples are given to illustrate the proposed method.

Subordination Results for p-Valent Meromorphic Functions Associated with a Linear Operator

In this paper, by making use of the Hadamard products, we obtain some subordination results for certain family of meromorphic functions defined by using a new linear operator.

On Approximation Properties of Modified Sázas-Mirakyan Operators via Jain Operators

In the present manuscript, we propose the modification of Jain operators which the generalization of Szasz-Mirakyan operators. These new class operators are linear positive operators of discrete type depending on a real parameters. We give theorem of degree of approximation and the Voronovskaya asymptotic formula.