On K-groups of operator algebras on HD n spaces and QD n spaces

We obtain the K-groups of the operator ideals contained in the class of Riesz operators. And based on the results, we calculate the K-groups of the operator algebras on HD nspaces and QDn spaces.     

Completeness of quasi-normed symmetric operator spaces

We show that (generalized) Calkin correspondence between quasi-normed symmetric sequence spaces and symmetrically quasi-normed ideals of compact operators on an infinite-dimensional Hilbert space preserves completeness. We also establish a semifinite version of this result.     

Isometries of certain operator spaces

Let and be Banach spaces, and be the spaces of bounded linear operators from into In this paper we give full characterization of isometric onto operators of for a certain class of Banach spaces, that includes We also characterize the isometric onto operators of and the compact operators on Furthermore, the multiplicative isometric onto operators of , when multiplication on is taken to be the Schur product, are characterized.

     

The dual of noncommutative Lorentz spaces

We prove some noncommutative analogues of the classical results about dual spaces of the classical Lorentz spaces.     

Superposition operator in Sobolev spaces on domains

For an arbitrary open set we characterize all functions on the real line such that for all . New element in the proof is based on Maz'ya's capacitary criterion for the imbedding .     

Crossed products of operator systems

In this paper we introduce the crossed product construction for a discrete group action on an operator system. In analogy to the work of E. Katsoulis and C. Ramsey, we describe three canonical crossed products arising from such a dynamical system. We describe how these crossed product constructions behave under G-equivariant maps, tensor products, and the canonical $C^{?}$-covers. We show that hyperrigidity is preserved under two of the three crossed products. Finally, using A. Kavruk's notion of an operator system that detects $C^{?}$-nuclearity, we give a negative answer to a question on operator algebra crossed products posed by Katsoulis and Ramsey.     

Maps preserving peripheral spectrum of generalized Jordan products of operators

Let X1 and X2 be complex Banach spaces with dimension at least three, A1 and A2 be standard operator algebras on X1 and X2, respectively. For k ≥ 2, let(i1, i2,..., im) be a finite sequence such that {i1, i2,..., im} = {1, 2,..., k} and assume that at least one of the terms in(i1,..., im) appears exactly once. Define the generalized Jordan product T1 o T2 o ··· o Tk= Ti1Ti2··· Tim+ Tim··· Ti2Ti1 on elements in Ai. This includes the usual Jordan product A1A2 + A2A1, and the Jordan triple A1 A2 A3 + A3 A2 A1. Let Φ : A1 → A2 be a map with range containing all operators of rank at most three. It is shown that Φ satisfies that σπ(Φ(A1) o ··· o Φ(Ak)) = σπ(A1 o ··· o Ak) for all A1,..., Ak,where σπ(A) stands for the peripheral spectrum of A, if and only if Φ is a Jordan isomorphism multiplied by an m-th root of unity.     

Applications of operator spaces to abstract harmonic analysis

We give a survey of how the relatively young theory of operator spaces has led to a deeper understanding of the Fourier algebra of a locally compact group (and of related algebras).     

Kantorovich算子在Ba空间中的逼近

利用Ditzian-Totik光滑模,研究了Kantorovich算子在Ba空间中的逼近.得到逼近的正定理与等价定理.所得结果改进,推广和统一了一些作者的结果.     

Weak exactness for dual operator spaces

We introduce a new tensor product and study the weak^∗ condition C^′, which is also called weak^∗ exactness, for dual operator spaces. Our definition of weak^∗ condition C^′ is equivalent to Kirchberg's notion of weak exactness in the case of von Neumann algebras. We also study the connection of weak^∗ exact W^∗-TROs with their linking von Neumann algebras and study the structure of exact (respectively, nuclear) W^∗-TROs.     

Strong 3-commutativity preserving maps on standard operator algebras

Let X be a Banach space of dimension ≥ 2 over the real or complex field F and A a standard operator algebra in B(X). A map Φ :A →A is said to be strong 3-commutativity preserving if [Φ(A), Φ(B)]3 = [A,B]3 for all A,B∈ A, where[A,B]3 is the 3-commutator of A,B defined by[A, B]3 = [[[A, B],B],B] with [A,B] = AB-BA. The main result in this paper is shown that.,if Φ is a surjective map on A, then Φ is strong 3-commutativity preserving if and only if there exist a functional h : A →F and a scalar λ∈ F with λ~4 = 1 such that Φ(A)=λ A+h(A)I for all A ∈ A.     

Bounded Toeplitz products on Bergman spaces of the unit ball

We consider the question for which square integrable analytic functions f and g on the unit ball the densely defined products are bounded on the weighted Bergman spaces. We prove results analogous to those we obtained in the setting of the unit disk and the polydisk.     

Riesz potential operator in continual variable exponents Herz spaces

We find conditions on the variable parameters $p\left(x\right),q\left(t\right)$ and $\alpha \left(t\right)$, defining the Herz space $H^{p(·),q(·),\alpha (·)}\left({\mathbb{R}}^n\right)$, for the validity of Sobolev type theorem for the Riesz potential operator to be bounded within the frameworks of such variable exponents Herz spaces. We deal with a “continual” version of Herz spaces (which coincides with the “discrete” one when q is constant).     

Lindelöf powers and products of function spaces

We give criteria for finite and countable powers of a space similar to the Michael line being Lindelöf. As applications, we give examples related to Lindelöf property in products of spaces of Michael line type and in products of spaces of continuous functions on separable -compact spaces.

     

Maximal operator on variable Lebesgue spaces for almost monotone radial exponent

We study general Lebesgue spaces with variable exponent p. It is known that the classes L and N of functions p are such that the Hardy-Littlewood maximal operator is bounded on them provided p∈L∩P. The class L governs local properties of p and N governs the behavior of p at infinity.In this paper we focus on the properties of p near infinity. We extend the class N to a collection D of functions p such that the Hardy-Littlewood maximal operator is bounded on the corresponding variable Lebesgue spaces provided p∈L∩D and the class D is essentially larger than N.Moreover, the condition p∈D is quite easily verifiable in the practice.     

Boundedness of the Cesàro operator on mixed norm spaces

In this note, the boundedness of the Cesàro operator on mixed norm space , , is proved.