一类可逆的弱次正规算子的幂

It is known that the square of a ω- hyponormal operator is also ω- hyponormal. In this note it is showed that there exists an invertible operator which is not log-hyponormal but its integer powers are all ω-hyponormal.     

PROPERTIES OF p-ω-HYPONORMAL OPERATORS

The approximate point spectrum properties of p-ω-hyponormal operators are given and proved. In faet, it is a generalization of approximate point speetrum properties of ω- hyponormal operators. The relation of spectra and numerical range of p-ω-hyponormal operators is obtained, On the other hand, for p-ω-hyponormal operators T,it is showed that if Y is normal,then T is also normal.     

Approximation by semigroup of spherical operators

This paper concerns about the approximation by a class of positive exponential type multiplier operators on the unit sphere Sn of the （n ＋ 1）- dimensional Euclidean space for n ≥2. We prove that such operators form a strongly continuous contraction semigroup of class （l0） and show the equivalence between the approximation errors of these operators and the K-functionals. We also give the saturation order and the saturation class of these operators. As examples, the rth Boolean of the generalized spherical Abel-Poisson operator ＋Vt^γ and the rth Boolean of the generalized spherical Weierstrass operator ＋Wt^k for integer r ≥ 1 and reals γ, k∈ （0, 1] have errors ｜｜＋r Vt^γ- f｜｜X ω^rγ（f, t^1/γ）X and ｜｜＋rWt^kf - f｜｜X ω^2rk（f, t^1/（2k））X for all f ∈ X and 0 ≤t ≤2π, where X is the Banach space of all continuous functions or all L^p integrable functions, 1 ≤p ≤＋∞, on S^n with norm ｜｜·｜｜X, and ω^s（f,t）X is the modulus of smoothness of degree s 〉 0 for f ∈X. Moreover, ＋r^Vt^γ and ＋rWt^k have the same saturation class if γ= 2k.     

Study on the generator f of the operators I n and I t

Define two operators In and It,the inner product operator In(g)(x) := j∈Zs(g,f(·-j))f(x-j) and the interpolation operator It(g)(x) := j∈Zs g(j)f(x-j),where f belongs to some space and integer s 1.We call f the generator of the operators In and It.It is well known that there are many results on operators In and It.But there remain some important problems to be further explored.For application we first need to find the available generators (that can recover polynomials as It(p) = p or In(p) = p,p ∈Πm-1) for constructing the relative operators.In this paper,we focus on the available generator in the class of spline functions.We shall see that not all spline functions can be used to construct available generators.Fortunately,we do find a spline function in S,of degree m-1,where m is even and S is a class of splines.But for odd m the problem is still open.Results on spline functions in this paper are new.     

Reflexivity of Weighted Shifts

Let {β（n）}n be a sequence of positive numbers such that β（0） = 1 and let 1 ≤ p 〈 ≤. We will investigate the reflexivity of all integer powers of the multiplication operator on the Banach spaces of formal Laurent series, L^P（β）.     

NON-STANDARD COMMUTATORS FOR ROUGH OSCILLATORY SINGULAR INTEGRALS

In this paper, the author studies a class of non-standard commutators with higher order remainders for oscillatory singular integral operators with phases more general than polynomials. For 1 〈 p 〈 ∞, the L^p-boundedness of such operators are obtained provided that their kernels belong to the spaces L^q（s^n-1） for some q 〉 1.     

L~p CONTINUITY OF HRMANDER SYMBOL OPERATORS OpS_（0,0） ~m AND NUMERICAL ALGORITHM

If we use Littlewood-Paley decomposition, there is no pseudo-orthogonality for Ho¨rmander symbol operators OpS m 0 , 0 , which is different to the case S m ρ,δ (0 ≤δ < ρ≤ 1). In this paper, we use a special numerical algorithm based on wavelets to study the L p continuity of non infinite smooth operators OpS m 0 , 0 ; in fact, we apply first special wavelets to symbol to get special basic operators, then we regroup all the special basic operators at given scale and prove that such scale operator’s continuity decreases very fast, we sum such scale operators and a symbol operator can be approached by very good compact operators. By correlation of basic operators, we get very exact pseudo-orthogonality and also L 2 → L 2 continuity for scale operators. By considering the influence region of scale operator, we get H 1 (= F 0 , 2 1 ) → L 1 continuity and L ∞→ BMO continuity. By interpolation theorem, we get also L p (= F 0 , 2 p ) → L p continuity for 1 < p < ∞ . Our results are sharp for F 0 , 2 p → L p continuity when 1 ≤ p ≤ 2, that is to say, we find out the exact order of derivations for which the symbols can ensure the resulting operators to be bounded on these spaces.     

Existence and Multiplicity Results for a Degenerate Elliptic Equation

The goal of this paper is to study the multiplicity result of positive solutions of a class of degenerate elliptic equations. On the basis of the mountain pass theorems and the sub- and supersolutions argument for p-Laplacian operators, under suitable conditions on the nonlinearity f（x, s）, we show the following problem：-△pu=λu^α-a（x）u^q in Ω,u│δΩ=0 possesses at least two positive solutions for large λ, where Ω is a bounded open subset of R^N, N ≥ 2, with C^2 boundary, λ is a positive parameter, Ap is the p-Laplacian operator with p 〉 1, α, q are given constants such that p - 1 〈α 〈 q, and a（x） is a continuous positive function in Ω^-.     

Spectrum of a class of fourth order left-definite differential operators

The spectrum of a class of fourth order left-definite differential operators is studied. By using the theory of indefinite differential operators in Krein space and the relationship between left-definite and right-definite operators, the following conclusions are obtained： if a fourth order differential operator with a self-adjoint boundary condition that is left-definite and right-indefinite, then all its eigenvalues are real, and there exist countably infinitely many positive and negative eigenvalues which are unbounded from below and above, have no finite cluster point and can be indexed to satisfy the inequality …≤λ-2≤λ-1≤λ-0〈0〈λ0≤λ1≤λ2≤…     

The ρ-variation of the Hermitian Riesz transform

In this paper we prove the behaviour in weighted Lp spaces of the oscillation and variation of the Hilbert transform and the Riesz transform associated with the Hermite operator of dimension 1. We prove that this operator maps LP（R, w（x）dx） into itself when w is a weight in the Ap class for 1 〈 p 〈 ∞. For p = 1 we get weak type for the A1 class. Weighted estimated are also obtained in the extreme case p = ∞.     

关于p-弱亚正规算子的一个注记

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关于(s,p)-w-亚正常算子类

本文在W-亚正常算子类的基础上，引入（s，p）-w-亚正常算子类，进而讨论了该类算子的特征，包含关系，对一类特殊的该类算子还考虑了其平方性质．     

p-ω-亚正规算子的一些性质

In this paper,let T be a bounded linear operator on a complex Hilbert H.We give and prove that every p-w-hyponormal operator has Bishop's property(β)and spectral properties;Quasi-similar p-w-hyponormal operators have equal spectra and equal essential spectra.Finally,for p-w-hyponormal operators,we give a kind of proof of its normality by use of properties of partial isometry.     

Full Fourier-Bessel transform and the algebra of singular pseudodifferential operators

The one-dimensional full Fourier-Bessel transform was introduced by I.A. Kipriyanov and V.V. Katrakhov on the basis of even and odd small (normalized) Bessel functions. We introduce a mixed full Fourier-Bessel transform and prove an inversion formula for it. Singular pseudodifferential operators are introduced on the basis of the mixed full Fourier-Bessel transform. This class of operators includes linear differential operators in which the singular Bessel operator and its (integer) powers or the derivative (only of the first order) of powers of the Bessel operator act in one of the directions. We suggest a method for constructing the asymptotic expansion of a product of such operators. We present the form of the adjoint singular pseudodifferential operator and show that the constructed algebra is, in a sense, a *-algebra.     

关于wF(p,r,q)算子类

作为wA(p,r)算子类的一个推广,该文介绍了一类更广泛的算子类即 wF(p,r,q)算子类,它包含A(p,r)类而含于F(p,r,q)类之中, 进而考虑了该类算子的特征, 包含关系, 正规性和幂性质等等.     

关于B-凸空间及其上黎斯算子West分解

给出 Ｂａｎａｃｈ空间列｛Ｘｉ｝ｉ＝１∞的 ｌｐ乘积Ｂ－凸的特征刻划, 证明Ｂ－凸空间上的每个黎斯算子可Ｗｅｓｔ分解,即分解成一个紧算子和一个拟幂 零算子的和．     

On congruences for the traces of powers of some matrices

In a series of recent papers, V.I. Arnold studied many questions concerning the statistics and dynamics of powers of elements in algebraic systems. In particular, on the basis of experimental data, he proposed an Euler-type congruence for the traces of powers of integer matrices as a conjecture. The proof of this conjecture was deduced from the author’s theorem (obtained at the end of 2004) on congruences for the traces of powers of elements in number fields. Recently, it turned out that there also exist other approaches to congruences for the traces of powers of integer matrices. In the present paper, the author’s results of 2004 are strengthened and a survey of their relations to number theory, theory of dynamical systems, combinatorics, and p-adic analysis is given. The main conclusion of this survey is that all approaches considered here ultimately reflect different points of view on a certain simple but important phenomenon in mathematics.     

The Density Problem for Unbounded Bergman Operators

The unbounded Bergman operator, the operator of multiplication by on an unbounded open subset of the plane, is considered. We give a complete answer regarding the density problem of unbounded Bergman operators in terms of its equivalence to the problem of bounded point evaluations for the Bergman spaces. Using this equivalence and the notion of Wiener capacity, we obtain simple geometric conditions that classify almost those open subsets of the plane for which the corresponding Bergman operators are densely defined. With the aid of an analytic approach, we are also able to give condition for a large collection of open subsets of the plane for which all the positive integer powers of the corresponding Bergman operators are densely defined. Submitted: December 14, 2001? Revised: January 14, 2001.