Researches on the Spaces of Analytic and Harmonic Functions and Their Operators

1. 研究了开单位圆盘上有界解析函数在H∞空间上的复合算子的性质。T. Hosokawa， K. J. Izuchi等人在H∞上刻画了加权复合算子集的拓扑结构，并发表了一篇论文。作者继续研究了组合算子在H∞空间上的线性组合，完整地刻画了组合算子的紧性，并估计了系数为正情况下的本质范数。Izuchi和他扩展了这些结果，得到了基本范数的估计，使得系数的实部为正，并给出了结果。2. K.J. Izuchi等研究了开单位盘上有界调和函数空间上的hankel型算子的紧性和完全连续性。这些算子与紧一致代数、Dunford-Pettis性质和Bourgain代数有关。此外，作者还研究了Hardy和Bergman空间的情况，并在RIMS联合研究“解析函数空间及其算子”（2006年6月21日（星期三）- 23日（星期五），京都大学数学科学研究所）上作了报告。本次会议由作者与本项目有关的人员申请并通过。3. T. Hosokawa等在算子拓扑中的Bloch空间上研究了复合算子空间的拓扑结构，并发表了论文。进一步考虑了两个复合算子在Bloch和小Bloch空间上的差的有界性和紧性，证明了差在小Bloch空间上的弱紧性等价于紧性。这篇论文也被接受。4. Hibschweiler和Portnoy在Hardy和加权Bergman空间上定义了复合算子和微分算子的乘积，并利用carleson型测度研究了加权Bergman空间之间的有界性和紧性。但是这样的加权Bergman空间在描述复合算子和微分算子的积的有界性和紧性时不包括Hardy空间的情况。作者对这一问题进行了研究并发表。少

1. We have investigated properties of composition operators on the space H∞ of bounded analytic functions on the open unit disk. T. Hosokawa, K. J. Izuchi and the author characterized the topological structure of the set of weighted composition operators on H∞ and published a paper. The author has continued to study linear combinations of composition operators on the space H∞ and completely characterized the compactness and estimated the essential norms in the case that coefficients are positive. And Izuchi and he extended these results and obtained the estimation of the essential norms such that real parts of coefficients are positive and submitted results. 2. K.J. Izuchi and the author characterized the compactness and complete continuity of Hankel-type operators on the space of bounded harmonic functions on the open unit disk and published. These operators related to tight uniform algebras, the Dunford-Pettis property, and Bourgain algebras. Moreover the author studied the cases of … More Hardy and Bergman spaces and had a talk at RIMS Joint Research "Analytic Function Spaces and Their Operators" (June 21(Wed) - 23(Fri), 2006,Kyoto University Research Institute for Mathematical Science). This conference was applied by the author with the relation ship to this project and accepted. 3. T. Hosokawa and the author studied the topological structure of the space of composition operators on the Bloch space in the operator topology and published. Furthermore we considered the boundedness and the compactness of the differences of two composition operators on the Bloch and the little Bloch spaces and proved that the weak compactness of the differences on the little Bloch space is equivalent to the compactness. This paper also is accepted. 4. Hibschweiler and Portnoy defined the products of composition and differentiation operators on Hardy and weighted Bergman spaces and investigated the boundedness and the compactness between weighted Bergman spaces using the Carleson-type measures. But such weighted Bergman spaces would not include the Hardy space case in the characterization of boundedness and compactness of the products of composition and differentiation operators. The author studied this problem and published. Less