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.     

Certain free products of operator spaces

Certain free products are introduced for operator spaces and dual operator spaces. It is shown that the free product of operator spaces does not preserve the injectivity. The linking C*-algebra of the full free product of two ternary rings of operators (simply, TRO's) is *-isomorphic to the full free product of the linking C*-algebras of the two TRO's. The operator space-reduced free product of the preduals of von Neumann algebras agrees with the predual of the reduced free product of the von Neumann algebras. Each of two operator spaces can be embedded completely isometrically into the reduced free product of the operator spaces. Finally, an example is presented to show that the C*-algebra-reduced free product of two C*-algebras may be contractively isomorphic to a proper subspace of their reduced free product as operator 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 .     

Isometries of certain operator algebras

To a given basis on an -dimensional Hilbert space , we associate the algebra of all linear operators on having every as an eigenvector. So, is commutative, semisimple, and -dimensional. Given two algebras of this type, and , there is a natural algebraic isomorphism of and . We study the question: When does preserve the operator norm?

     

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.     

Isometries on and monotone functions

We study necessary and sufficient conditions on a bounded operator T defined on the Hilbert space to be an isometry and show that, under suitable hypotheses, it suffices to restrict T to a smaller class of functions (e.g., if , to the cone of positive and decreasing functions). We also consider the problem of characterizing the sets for which the orthogonal projection of the operator T on is also an isometry. Finally, we illustrate our results with several examples involving classical operators on different settings.     

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 type estimates on certain Hardy spaces for smooth cone type multipliers

Let ?n∈C^∞(Rd?{0}) be a non-radial homogeneous distance function of degree n∈N satisfying ?n(tξ)=tn?n(ξ). For f∈S(Rd+1) and δ>0, we consider convolution operator associated with the smooth cone type multipliers defined by

     

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).     

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.

     

分数次Orlicz极大算子在齐型空间中的局部加权端点估计

利用齐型空间中的覆盖引理及其有界区域的二进方体分解得到了分数次Orlicz极大算子在齐型空间（X,d,μ）中的有界区域Ω上的局部加权端点估计．该工作为分数次积分交换子[b,Iα】在欧式空间R^n中的有界区域上的加权端点弱型估计推广到齐型空间奠定了基础．     

The continuous spectrum of the Dirac operator on noncompact Riemannian symmetric spaces of rank one

The continuous spectrum of the Dirac operator on the complex, quaternionic, and octonionic hyperbolic spaces is calculated using representation theory. It is proved that , except for the complex hyperbolic spaces with even, where .

     

A simple characterization of weighted Sobolev spaces with bounded multiplication operator

In this paper we give a simple characterization of weighted Sobolev spaces (with piecewise monotone weights) such that the multiplication operator is bounded: it is bounded if and only if the support of μ₀ is large enough. We also prove some basic properties of the appropriate weighted Sobolev spaces. To have bounded multiplication operator has important consequences in Approximation Theory: it implies the uniform bound of the zeros of the corresponding Sobolev orthogonal polynomials, and this fact allows to obtain the asymptotic behavior of Sobolev orthogonal polynomials.     

On the Libera operator

We show that the Libera operator, L, on some spaces of analytic functions is a continuous extension of the conjugate of the Cesàro operator. Results on L acting on various spaces are obtained. In particular, L maps the Bloch space into BMOA. We also prove some results on the best approximation by polynomials in Hardy and Bergman spaces.     

Area operator on Bergman spaces

We characterize the non-negative measuresμon the unit disk D for which the area operator Aμis bounded from Bergman space Aαp to Lq ((?)D).     

Smooth points of certain operator spaces

Let B(H) denote the Banach space of all bounded linear operators on a separable, infinite dimensional Hilbert space H, and let C be the ideal of compact operators on H. It is shown that the unit ball of the Calkin algebra B(H)/C has no smooth points. We use this to determine the smooth points of the unit ball of B(H).