A Maurey type result for operator spaces

The little Grothendieck theorem for Banach spaces says that every bounded linear operator between C(K) and ?₂ is 2-summing. However, it is shown in [M. Junge, Embedding of the operator space OH and the logarithmic ‘little Grothendieck inequality’, Invent. Math. 161 (2) (2005) 225-286] that the operator space analogue fails. Not every cb-map is completely 2-summing. In this paper, we show an operator space analogue of Maurey's theorem: every cb-map is (q,cb)-summing for any q>2 and hence admits a factorization ‖v(x)‖?c(q)‖vcb_‖‖axbq_‖ with a,b in the unit ball of the Schatten class S2q.     

Matrix regular operator space and operator system

We establish a relationship between Schreiner's matrix regular operator space and Werner's (nonunital) operator system. For a matrix ordered operator space V with complete norm, we show that V is completely isomorphic and complete order isomorphic to a matrix regular operator space if and only if both V and its dual space V^∗ are (nonunital) operator systems.     

Stable isomorphism of dual operator spaces

We prove that two dual operator spaces X and Y are stably isomorphic if and only if there exist completely isometric normal representations ? and ψ of X and Y, respectively, and ternary rings of operators M₁, M₂ such that and . We prove that this is equivalent to certain canonical dual operator algebras associated with the operator spaces being stably isomorphic. We apply these operator space results to prove that certain dual operator algebras are stably isomorphic if and only if they are isomorphic. Consequently, we obtain that certain complex domains are biholomorphically equivalent if and only if their algebras of bounded analytic functions are Morita equivalent in our sense. Finally, we provide examples motivated by the theory of CSL algebras.     

On the approximating local lifting property for operator spaces

In this paper, we define and study the approximately local lifting property for operator spaces. We show that an operator space V has the approximately local lifting property if and only if V^∗ is injective. This implies that an operator space V has the approximately local lifting property if and only if it has the local lifting property.     

Inequalities and separation for the Laplace-Beltrami differential operator in Hilbert spaces

In this paper we have studied the separation for the Laplace-Beltrami differential operator of the form

     

Properties of operator stable distributions in infinite-dimensional Banach spaces

The results of this paper concern the connections between different characterizations of operator stable distributions in infinite-dimensional separable Banach spaces. Such assertions are closely related with the strong and uniform operator topology. In particular, the new motions of strongly and uniformlyG-stable distributions are discussed.     

An approximation theorem for nuclear operator systems

We prove that an operator system S is nuclear in the category of operator systems if and only if there exist nets of unital completely positive maps φλ:S→Mnλ and ψλ:Mnλ→S such that ψλ°φλ converges to idS in the point-norm topology. Our proof is independent of the Choi-Effros-Kirchberg characterization of nuclear C^?-algebras and yields this characterization as a corollary. We give an explicit example of a nuclear operator system that is not completely order isomorphic to a unital C^?-algebra.     

The boundedness of maximal Bochner-Riesz operator and maximal commutator on Morrey type spaces

The boundedness of maximal Bochner-Riesz operator Bδ* and that of maximal commutator Bδb,* generated by this operator and Lipschitz function on the classical Morrey space and generalized Morrey space are established.     

加权Morrey空间上分数次极大算子的双权不等式

Ye与Wang研究了Hardy-Littlewood极算子在加权Morrey空间的双权不等式.该文将Ye与Wang的结果拓展到分数次极大算子,此外也得到了Ap型的充分条件.     

Perturbation of spectra of operator matrices and local spectral theory

For A∈L(X), B∈L(Y) and C∈L(Y,X) we denote by MC the operator defined on X⊕Y by . In this article, we study defect set DΣ=(Σ(A)∪Σ(B))?Σ(MC) for different spectra including the spectrum, the essential spectrum, Weyl spectrum and the approximate point spectrum. We then apply the obtained results to the stability of such spectra (DΣ=∅) and the classes of operators C for which stability holds of MC using local spectral theory.     

Continuity of the maximal operator in Sobolev spaces

We establish the continuity of the Hardy-Littlewood maximal operator on Sobolev spaces , . As an auxiliary tool we prove an explicit formula for the derivative of the maximal function.

     

Polynomials in operator space theory

The aim of this article is to start a metric theory of homogeneous polynomials in the category of operator spaces. For this purpose we take advantage of the basic fact that the space P^m(E)$P^m(E)$ of all m-homogeneous polynomials on a vector space E can be identified with the algebraic dual of the m  -th symmetric tensor product ⊗^m,sE${\otimes }^{m,s}E$. Given an operator space V, we study several different types of completely bounded polynomials on V   which form the operator space duals of ⊗^m,sV${\otimes }^{m,s}V$ endowed with related operator structures. Of special interest are what we call Haagerup, Kronecker, and Schur polynomials – polynomials associated with different types of matrix products.     

On maximal functions in Orlicz spaces

Let and be the functions having the representations and , where is a positive continuous function such that and is quasi-increasing. Then the maximal function is a function in Orlicz space for all if and only if there exists a positive constant such that for all .

     

Approximating composition operator norms on the Dirichlet space

Let φ be any univalent self-map of the unit disk D whose image Ω≡φ(D) is compactly contained in D. We provide a method for approximating the norm of the composition operator Cφ on the Dirichlet space to any desired degree of accuracy. The approximation uses a special basis which is orthogonal in both the Bergman space on the disk and the Bergman space on Ω.     

Topological vector spaces admissible in economic equilibrium theory

In models of economic equilibrium in markets with infinitely many commodities, the commodity space is an ordered topological vector space endowed with additional structure. In the present paper, we consider ordered topological vector spaces which are admissible (for equilibrium analysis) in the sense that every economy which is reasonably well behaved posesses an equilibrium. It turns out that this condition may be characterized in terms of topology and order. This characterization implies that the commodity space has the structure of a Kakutani space.     

Minimal bases and maximal nonbases in additive number theory

An asymptotic basis $\text{A}$ of order h is minimal if no proper subset of $\text{A}$ is an asymptotic basis of order h. Examples are constructed of minimal asymptotic bases, and also of an asymptotic basis of order two no subset of which is minimal.If $\text{B}$ is a set of nonnegative integers which is not a basis (resp. asymptotic basis) of order h, but such that every proper superset of $\text{B}$ is a basis (resp. asymptotic basis) of order h, then $\text{B}$ is a maximal nonbasis (resp. maximal asymptotic nonbasis) of order h. Examples of such sets are constructed, and it is proved that every set not a basis of order h is a subset of a maximal nonbasis of order h.     

Riesz product spaces and representation theory

Let {E _i:i∈I} be a family of Archimedean Riesz spaces. The Riesz product space is denoted by ∏ _i∈I E_i. The main result in this paper is the following conclusion: There exists a completely regular Hausdorff spaceX such that ∏ _i∈I E_i is Riesz isomorphic toC(X) if and only if for everyi∈I there exists a completely regular Hausdorff spaceX _i such thatE _i is Riesz isomorphic toC(X _i). Supported by the National Natural Science Foundation of China     

Complexifications of symmetric spaces and Jordan theory

Generalizing Hermitian and pseudo-Hermitian spaces, we define twisted complex symmetric spaces, and we show that they correspond to an algebraic object called Hermitian Jordan triple products. The main topic of this work is to investigate the class of real forms of twisted complex symmetric spaces, called the category of symmetric spaces with twist. We show that this category is equivalent to the category of all real Jordan triple systems, and we can use a work of B.O. Makarevic in order to classify the irreducible spaces. The classification shows that most irreducible symmetric spaces have exactly one twisted complexification. This leads to open problems concerning the relation of Jordan and Lie triple systems.