On a class of second order partial differential operators of non-principal type

Acta Mathematica Sinica, English Series -     

ON THE RATES OF CONVERGENCE FOR PROBABILITY TYPE OPERATORS SEQUENCE

Abstract. In this paper, the rates of convergence for some probability type operators sequence are obtained. The quantitative Poisson type limit theorem is established as an application.     

On the exact WKB analysis of higher order simple-pole type operators

Higher order simple-pole type operators, that is, higher order linear ordinary differential operators with a large parameter η whose coefficients have simple poles at the origin, are discussed from the viewpoint of the exact WKB analysis. Making use of the technique of microdifferential operators, we clarify the singularity structure of the Borel transform of their WKB solutions.     

Second order differential operators with Feller-Wentzell type boundary conditions

In this paper, we study a basic generation problem concerning the second order differential operator in the space C[0,1] of complex continuous functions equipped with Feller-Wentzell type boundary conditions, which originates from the work of Feller [W. Feller, The parabolic differential equations and the associated semi-groups of transformations, Ann. of Math. (2) 55 (1952) 468-519]. We prove successfully that the operator, under suitable assumptions, generates a strongly continuous cosine function on C[0,1] (or on a subspace of C[0,1]), by means of an operator matrix analysis combined with perturbation, approximation, and similarity techniques.     

Borel type of the convergence set of a sequence of bounded linear operators in a Banach space

The connection between the Borel type of the convergence set of a sequence of bounded linear operators, acting from one Banach space into another, and the linear-topological properties of this second space is studied.Translated from Teoriya Funktsii, Funktsional'nyi Analiz i Ikh Prilozheniya, No. 50, pp. 90–98, 1988.     

On the m order difference sequence space of generalized weighted mean and compact operators

In this article, using generalized weighted mean and difference matrix of order m, we introduce the paranormed sequence space ?(u, v, p; Δ^(m)), which consist of the sequences whose generalized weighted Δ^(m)-difference means are in the linear space ?(p) defined by I.J. Maddox. Also, we determine the basis of this space and compute its α-, β- and γ-duals. Further, we give the characterization of the classes of matrix mappings from ?(u, v, p, Δ^(m)) to ?_∞, c and c₀. Finally, we apply the Hausdorff measure of noncompacness to characterize some classes of compact operators given by matrices on the space ?_p(u, v, Δ^(m))(1 ≤ p < ∞).     

On a third‐order Newton‐type method free of bilinear operators

This paper is devoted to the study of a third‐order Newton‐type method. The method is free of bilinear operators, which constitutes the main limitation of the classical third‐order iterative schemes. First, a global convergence theorem in the real case is presented. Second, a semilocal convergence theorem and some examples are analyzed, including quadratic equations and integral equations. Finally, an approximation using divided differences is proposed and used for the approximation of boundary‐value problems. Copyright © 2009 John Wiley & Sons, Ltd.     

Semi-Fredholm operators and sequence conditions

A typical result of this paper is the following: A closed linear operator T between Banach spaces X and Y is lower semi-Fredholm iff it is a rk-(=relatively compact)surjection, i.e. for every bounded sequence (y_n) in Y there is a bounded sequence (x_n) in D(T) such that (y_n?Tx_n) is relatively compact in Y. The concept of c_o-(rk- and wk-)injections and surjections are introduced to characterize semi-Fredholm properties (in the usual or generalized sense). This is done also by new operator moduli.     

Higher order commutators for a class of rough operators

In this paper we study the (L ^p (u ^p ),L ^q (v ^q )) boundedness of the higher order commutatorsT _Ω,α,b ^m andM _Ω,α,b ^m formed by the fractional integral operatorT _Ω,α , the fractional maximal operatorM _Ω,α , and a functionb(x) in BMO(v), respectively. Our results imporve and extend the corresponding results obtained by Segovia and Torrea in 1993 [9]. The project was supported by NNSF of China.     

Approximation by a sequence of operators of the Srivastava‐Gupta type involving the Brenke polynomials with a certain parameter

We introduce a new sequence of linear positive operators by combining the Brenke polynomials and the Srivastava‐Gupta–type operators defined by Srivastava‐Gupta. obtain the moments of the operators and present some classical and statistical approximation properties by means of Korovkin results. Next, we estimate a global result, which includes the Voronovskaya‐type asymptotic formula, local approximation, error estimation in terms of weighted modulus of continuity, and for functions in a Lipschitz‐type space. Lastly, we estimate the rate of approximation for functions with derivatives of bounded variation.     

Spectral order of operators and range projections

We study the effect algebra (i.e. the positive part of the unit ball of an operator algebra) and its relation to the projection lattice from the perspective of the spectral order. A spectral orthomorphism is a map between effect algebras which preserves the spectral order and orthogonality of elements. We show that if the spectral orthomorphism preserves the multiples of the unit, then it is a restriction of a Jordan homomorphism between the corresponding algebras. This is an optimal extension of the Dye's theorem on orthomorphisms of the projection lattices to larger structures containing the projections. Moreover, results on automatic countable additivity of spectral homomorphisms are proved. Further, we study the order properties of the range projection map, assigning to each positive contraction in a JBW algebra its range projection. It is proved that this map preserves infima of positive contractions in the spectral (respectively standard order) if, and only if, the projection lattice of the algebra in question is a modular lattice.     

Perturbation of regular operators and the order essential spectrum

For regular operators on a Banach lattice, we introduce and investigate two notions of order essential spectrum analogous to the essential spectrum and the Weyl spectrum for operators on Banach spaces. We also discuss related questions on the behaviour of the order spectrum under perturbation by r-compact operators.     

On operator order and chaotic operator order for two operators

In this paper we characterize operator order A?B?O and chaotic operator order log A?logB for positive and invertible operators A and B in terms of operator inequalities via the Furuta inequality and operator equalities due to the Douglas’s majorization and factorization. Related results are obtained, which include generalizations and characterizations of some well-known results.     

Spectral order for unbounded operators

The spectral order, a notion originated by Olson for bounded operators, is investigated here in the context of unbounded operators. Dissimilarities between bounded and unbounded cases are pointed out. New criteria for two operators to be comparable are supplied. A way of reducing the study of the spectral order to the case of bounded operators is proposed. Connections with essential selfadjointness are established. Integral inequalities for monotonically increasing functions are characterized in terms of distribution functions. Some illustrative examples are furnished.