THE SCHUR TEST OF COMPACT OPERATORS

Infinite matrix theory is an important branch of function analysis.Every linear operator on a complex separable infinite dimensional Hilbert space corresponds to an infinite matrix with respect a orthonormal base of the space,but not every infinite matrix corresponds to an operator.The classical Schur test provides an elegant and useful criterion for the boundedness of linear operators,which is considered a respectable mathematical accomplishment.In this paper,we prove the compact version of the Schur test.Moreover,we provide the Schur test for the Schatten class S_(2).It is worth noting that our main results can be applicable to the general matrix without limitation on non-negative numbers.We finally provide the Schur test for compact operators from l_(p) into l_(q).

CHARACTERIZATIONS OF COMPACT OPERATORS ON SOME EULER SPACES OF DIFFERENCE SEQUENCES OF ORDER m

In the past, several authors studied spaces of m-th order difference sequences, among them, H.Polat and F.Basar （[17]） defined the Euler spaces of m-th order difference sequences e r 0 （△ （ m ） ）, e r c （△ （ m ） ） and e r ∞ （△ （ m ） ） and characterized some classes of matrix transformations on them. In our paper, we add a new supplementary aspect to their research by characterizing classes of compact operators on those spaces. For that purpose, the spaces are treated as the matrix domains of a triangle in the classical sequence spaces c 0 , c and ∞ . The main tool for our characterizations is the Hausdorff measure of noncompactness.

Compact Operators on the Bergman Spaces of the Unit Ball and the Polydisk in C^n

Let Ω be the unit ball or the polydisk of Cnand L2a(Ω) the Bergman space. In this paper we prove that if S is a finite sum of finite products of Toeplitz operators on L2a( Ω), then S is compact if and only if the Berezin transform S(z) of S tends to zero as z→Ω.

Remarks on the Riesz Separation Property of the Linear Span of Positive Compact Operators

It is proven that there exists a Dedekind complete Banach lattice E such that the linear spans/f （E） and IV （E） of positive compact and positive weakly compact operators on E fails to possess the Riesz separation property.

Recent Results on Weakly Compact Operators on Banach Lattices

There have been many really positive results co ncerning the weakly compact operators on Banach lattices in terms of their order structure as well as in many respects. This paper will survey some known recent results in this area.

COMPACT TOEPLITZ OPERATORS ON A^P (φ)(1

Let φ be a normal function defined on [0, 1) and A^p(φ) Bergman space weighted with φ~p(|z|)/(1-|z|~2) for 1≤p<∞. The compactnesses of Toeplitz operaters on A^p(φ) are characterized by Carleson measures and operator algebra.

A NOTE OF THE SPECTRUM OF HOMOGENEOUS COMPACT OPERATORS

In this paper, we present some counterexamples which show that there is no theory on the spectrum of homogeneous compact operators which parallels the Riesz-Schauder theory on the spectrum of linear compact operators. These counterexamples also illustrate that it is impossible to study in a unified setting the Fucik spectrum of the Laplacian: -△w = au+ - bu- inΩand u = 0 on (?)Ω, as well as the spectrum of the p-Laplacian: -div(|(?)u| p-2(?)u) = λ|u|p-2u and u = 0 on (?)Ω.

Kernel Thesis Reproduction and Truncated Toeplitz Operators

Important operator characteristics (such boundedness or compactness) for particular classes of operators on particular reproducing kernel Hilbert spaces may be impacted by the behaviour of the operators on the reproducing kernels. These results have been shown for Toeplitz operators on the Paley-Wiener space, a reproducing kernel Hilbert space over C. Furthermore, we show how the norm of such an operator has no relation to the supremum of the norms of the pictures of the normalization reproducing kernels of the space. As a result, if this supremum is finite, the operator is implicitly bounded. To further demonstrate that the operator norm is not the same as the supremum of the norms of the pictures of the real normalized reproducing kernels, another example is also provided. We also set out a necessary and sufficient condition for the operators’ compactness in terms of their limiting function on the reproducing kernels.

COLLECTIVELY COMPACT COMPOSITION OPERATOR SEQUENCES BETWEEN BERGMAN AND HARDY SPACES

This paper studies the collective compactness of composition operator sequences between the Bergman and Hardy spaces. Some sufficient and necessary conditions involving the generalized Nevanlinna counting functions for composition operator sequences to be collectively compact between weighted Bergman spaces are given

COMPACTNESS FOR THE COMMUTATOR OF BOCHNER-RIESZ OPERATOR

Let α ∈0,(n-1)/2 and T~α be the Bochner-Riesz operator of order α. In this paper, for n = 2 and n ≥ 3, the compactness on Lebesgue spaces and Morrey spaces are considered for the commutator of Bochner-Riesz operator generated by CMO(R^n) function and T~α.

Composition Operators From B0 to QK and QK,0 Spaces

When φ is an analytic map of the unit disk D into itself,and X is a Banach space of analytic functions on D,define the composition operator Cφ by Cφ(f) = f οφ,for f ∈ X.In this paper,we study the boundedness and compactness of composition operators from the space B0 to QK and QK,0.

COMPOSITION OPERATORS ON HARDY-ORLICZ SPACES

Properties of composition operators induced by analytic self-maps on the unit disk of the complex plane in Hardy-Orlicz spaces are discussed. Results are concerned about boundedness, invertibility, compactness, Fredholm operators and spectra of composition operators.

L^p CONTINUITY OF HRMANDER 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.

GENERALIZED CESàRO OPERATORS ON DIRICHLET-TYPE SPACES

In this note,we introduce and study a new kind of generalized Cesaro operator,C_(μ),induced by a positive Borel measure μ on(0,1)between Dirichlet-type spaces.We characterize the measures μ for which Cμis bounded(compact)from one Dirichlet-type space,Da,into another one,D_(β).

Quantitative Characterizations of Bounded Compact Approximation Property by Semi-Fredholm Operators

Based on a characterization of upper semi-Fredholm operators due to A.Lebow and M.Schechter,we introduce and investigate a new quantity characterizing upper semi-Fredholm operators.This quantity and several well-known quantities are used to characterize bounded compact approximation property.Similarly,a new quantity characterizing lower semi-Fredholm operators is introduced,investigated and used to characterize the bounded compact approximation property for dual spaces.

A Compact Difference Scheme for Time-Space Fractional Nonlinear Diffusion-Wave Equations with Initial Singularity

In this paper,we present a linearized compact difference scheme for onedimensional time-space fractional nonlinear diffusion-wave equations with initial boundary value conditions.The initial singularity of the solution is considered,which often generates a singular source and increases the difficulty of numerically solving the equation.The Crank-Nicolson technique,combined with the midpoint formula and the second-order convolution quadrature formula,is used for the time discretization.To increase the spatial accuracy,a fourth-order compact difference approximation,which is constructed by two compact difference operators,is adopted for spatial discretization.Then,the unconditional stability and convergence of the proposed scheme are strictly established with superlinear convergence accuracy in time and fourth-order accuracy in space.Finally,numerical experiments are given to support our theoretical results.

Compact Composition Operators Mapping into the E0(p,q) Spaces

Composition operators are used to study the E0(p,q) spaces, which coincide with the space Qq,0 for p = 2 and the little Bloch space B0 for p > 0 and q > 1. The compactness of these operators is also considered. The criteria for these operators to be compact are given in terms of the Carlesou measure.

EXISTENCE AND UNIQUENESS OF POSITIVE SOLUTIONS TO A CLASS OF SEMILINEAR ELLIPTIC SYSTEMS

In this paper an existence and uniqueness theorem of positive solutions to a class of semilinear elliptic systems is proved. Also, a necessary condition for the existence of the positive solution is obtained. As the application of the main theorem, two examples are given.

Compactness of the commutators of homogeneous singular integral operators

Let TΩ be the singular integral operator with kernel Ω（x）/｜x｜n where is homogeneous of degree zero, integrable and has mean value zero on the unit sphere Sn-1. In this paper, by Fourier transform estimates, Littlewood-Paley theory and approximation, the authors prove that if Ω∈（lnL）2 （Sn- 1）, then the commutator generated by TΩ and CMO（Rn） function, and the corresponding discrete maximal operator, are compact on LP（Rn, ｜s｜γp） for p∈ （1, ∞） and γp ∈ （-1, p-l）

Compactness of Product Operators on the Bergman Space

In this paper, we study the compactness of the product of a composition operator with another one＇s adjoint on the Bergman space. Some necessary and sufficient conditions for such operators to be compact are given.