Weak limits of the extreme dual ball in JB*-triples

We study weak limits of the extreme points, ∂ _e (E ^* ₁), of the dual ball of a JB*-triple, E. We show that all such weak limits, except possibly the zero functional, are weak sequential limits and we discuss implications for the structure of E. Received: 9 April 2001     

On convergence on the boundary of the unit ball in dual space

In this paper some results that are known for extreme points of the unit ball in dual space are carried over to a more general case, namely to the case of the boundary of the ball ( B is the boundary of the unit ballB in the space dual toX if everyx X achieves its maximum value onB at some point of ). For example, it is established that if a set is bounded inX and countably compact in(X, ), then it is weakly compact inX.Translated fromMatematickeskie Zametki, Vol. 59, No. 5, pp. 753–758, May, 1996.     

A new Banach space with Valdivia dual unit ball

We give an example of a Banach space which admits no projectional resolution of the identity but whose dual unit ball in weak* topology is a Valdivia compact. This answers a question asked by M. Fabian, G. Godefroy and V. Zizler. Partially supported by Research grants GAUK 277/2001, GAUK 160/1999 and MSM 113200007.     

单位球Dirichlet空间上的交换对偶Toeplitz算子

研究对偶Toeplitz算子（半）交换时其符号的关系.通过对其符号的分解,借助多复变函数的有关理论,得到了单位球Dirichlet空间上以多重调和函数为符号的对偶Toeplitz算子Sψ与Sψ（半）交换的充要条件.     

Hardy space interpolation in the unit ball

Interpolating sequences for the spaces H^p(Bⁿ), 1 <-p < ∞, n≥2, are studied. In contrast to the case n = 1, they are not the same for all p, but Varopoulos' necessary condition for H^∞ interpolation turns out to be necessary for H^p interpolation, for any p≥ 1, and sufficient for the sequence to be a finite union of H¹ interpolating sequences. Sufficient conditions are given for H^p interpolation, npn/(n-1), which are intermediary between Varopoulos' condition and Berndtsson's sufficient condition for H^∞(Bⁿ) interpolation.     

Separate weak*-continuity of the triple product in dual real JB*-triples

We prove that, if E is a real JB*-triple having a predual then is the unique predual of E and the triple product on E is separately $\sigma (E,E_{*_{}})-$continuous. Received February 1, 1999; in final form March 29, 1999 / Published online May 8, 2000     

On quasi-convex mappings of order α in the unit ball of a complex Banach space

In this paper, a class of biholomorphic mappings named quasi-convex mapping of order α in the unit ball of a complex Banach space is introduced. When the Banach space is confined to Cn, we obtain the relation between this class of mappings and the convex mappings.Furthermore, the growth and covering theorems of this class of mappings are given on the unit ball of a complex Banach space Ⅹ. Finally, we get the second order terms coefficient estimations of the homogeneous expansion of quasi-convex mapping of order α defined on the polydisc in Cn and on the unit ball in a complex Banach space, respectively.     

On growth and covering theorems of quasi-convex mappings in the unit ball of a complex banach space

A class of biholomorphic mappings named “quasi-convex mapping” is introduced in the unit ball of a complex Banach space. It is proved that this class of mappings is a proper subset of the class of starlike mappings and contains the class of convex mappings properly, and it has the same growth and covering theorems as the convex mappings. Furthermore, when the Banach space is confined to ℂⁿ, the “quasi-convex mapping” is exactly the “quasi-convex mapping of type A” introduced by K. A. Roper and T. J. Suffridge.     

The centroid of a weak*-closed inner ideal in a JBW*-triple

It is shown that there exists a ^*-homomorphism from the continuous centroid L^b (A){\cal L}^b (A) of a JBW^*-triple A onto the continuous centroid L^b (J){\cal L}^b (J) of an arbitrary weak^*-closed inner ideal J in A.     

Quantified functional analysis: recapturing the dual unit ball

We introduce the weak approach structure for an arbitrary locally convex approach space and generalize the results from [1] about the weak approach structure of a normed space. Hereto we carefully develop the notion of a closed dual unit ball in an abstract setting (as a special kind of absolutely convex subset) because it is this kind of structure on the algebraic dual that induces, in a duality-compatible way, a locally convex approach structure on the original space.     

On ε quasi-convex mappings in the unit ball of a complex Banach space

In this paper, a new class of biholomorphic mappings named “ε quasi-convex mapping” is introduced in the unit ball of a complex Banach space. Meanwhile, the definition of ε-starlike mapping is generalized from ε∈[0,1] to ε∈[−1,1]. It is proved that the class of ε quasi-convex mappings is a proper subset of the class of starlike mappings and contains the class of ε starlike mappings properly for some ε∈[−1,0)∪(0,1]. We give a geometric explanation for ε-starlike mapping with ε∈[−1,1] and prove that the generalized Roper-Suffridge extension operator preserves the biholomorphic ε starlikeness on some domains in Banach spaces for ε∈[−1,1]. We also give some concrete examples of ε quasi-convex mappings or ε starlike mappings for ε∈[−1,1] in Banach spaces or Cn. Furthermore, some other properties of ε quasi-convex mapping or ε-starlike mapping are obtained. These results generalize the related works of some authors.     

Integral-type operators from a mixed norm space to a bloch-type space on the unit ball

Let $ \mathbb{B} $ \mathbb{B} be the unit ball in ℂ ⁿ and let H($ \mathbb{B} $ \mathbb{B} ) be the space of all holomorphic functions on $ \mathbb{B} $ \mathbb{B} . We introduce the following integral-type operator on H($ \mathbb{B} $ \mathbb{B} ):

On a new integral-type operator from the Bloch space to Bloch-type spaces on the unit ball

We introduce the following integral-type operator on the space H(B) of all holomorphic functions on the unit ball B⊂Cn

     

On real forms of JB*-triples

We introduce real JB^*-triples as real forms of (complex) JB^*-triples and give an algebraic characterization of surjective linear isometries between them. As main result we show: A bijective (not necessarily continuous) linear mapping between two real JB^*-triples is an isometry if and only if it commutes with the cube mappinga→a ³={aaa}. This generalizes a result of Dang for complex JB^*-triples. We also associate to every tripotent (i.e. fixed point of the cube mapping) and hence in particular to every extreme point of the unit ball in a real JB^*-triple numerical invariants that are respected by surjective linear isometries.     

Hankel operators on the Bergman space of the unit ball

We characterize the bounded holomorphic functions in the unit ball of for which the operator is compact. For the result was obtained by Axler and Gorkin in 1988 and by Zheng in 1989.

     

Riemann-Hilbert problems for poly-Hardy space on the unit ball

In this paper, we focus on a Riemann–Hilbert boundary value problem (BVP) with a constant coefficients for the poly-Hardy space on the real unit ball in higher dimensions. We first discuss the boundary behaviour of functions in the poly-Hardy class. Then we construct the Schwarz kernel and the higher order Schwarz operator to study Riemann–Hilbert BVPs over the unit ball for the poly-Hardy class. Finally, we obtain explicit integral expressions for their solutions. As a special case, monogenic signals as elements in the Hardy space over the unit sphere will be reconstructed in the case of boundary data given in terms of functions having values in a Clifford subalgebra. Such monogenic signals represent the generalization of analytic signals as elements of the Hardy space over the unit circle of the complex plane.