Researches on Properties of Banach Spaces of Analytic Functions and Their Operators

根据我们的研究计划，我们研究了解析函数的空间及其算子，主要是“复合算子”B.MacCluer， S.Ohno和R.Zhao在一致算子拓扑H^∞上刻画了复合算子拓扑空间的分量和孤立点，以及两个复合算子的紧致差。借助这些结果，我们证明了复合算子空间中的一个分量一般不是与给定的复合算子有紧算子之差的所有复合算子的集合我们认为加权复合算子是复合算子和乘法算子的推广。(1) S.Ohno， K.Stroethoff和R.Zhao刻画了单位圆盘上解析函数的Bloch型空间（包括Lipschitz空间和Bloch空间）之间加权复合算子有界或紧致的充分必要条件。连续地，我们将考虑小bloch型空间的情况。(2) S.Ohno在H^∞和Bloch空间之间的加权复合算子。此外，S.Ohno在H^2和Bloch空间之间考虑它们。我们认为，在这种情况下的研究，可能与描述在H^2上的所有复合算子的空间中孤立的复合算子的问题（桑德伯格和夏皮罗的问题）有关。(3) S.Ohno和H. takagi研究了盘代数和H^∞上的加权复合算子。对于这些算子，我们证明了紧性、弱紧性和完全连续性的等价性。此外，我们还给出了加权复合算子具有闭值域或为Fredholm算子的充要条件s.o ohno定义了由复合算子诱导的de Branges-Rovnyak空间，并简化了由D.Sarason引起的一般情况的初等结果。我们可以考虑乘子和不变子空间的问题。

According to our researches' plan, we have investigated the spaces of analytic functions and their operators, mainly, " composition operators".[1] B.MacCluer, S.Ohno and R.Zhao characterized components and isolated points of the topological space of composition operators on H^∞ in the uniform operator topology and also compact differences of two composition operators. With the aid of these results, we showed that a component in the space of composition operators is not in general the set of all composition operators that differ from the given one by a compact operator.[2] We have considered weighted composition operators as the generalization of composition and multiplication operators.(1) S.Ohno, K.Stroethoff and R.Zhao characterized the necessary and sufficient conditions for weighted composition operators to be bounded or compact between Bloch-type spaces of analytic functions on the unit disk including Lipschitz spaces and Bloch spaces. Continuously, we would consider the case of small Bloch-type spaces.(2) S.Ohno characterized weighted composition operators between H^∞ and Bloch space. Moreover, S.Ohno considered them between H^2 and Bloch space. The investigation in this situation, we suppose, might have some relationship to the problem (Sundberg and Shapiro's problem) of characterizing composition operators that are isolated in the space of all composition operator on H^2.(3) S.Ohno and H.Takagi studied weighted composition operators on the disk algebra and H^∞. For these operators, we proved the equivalence of the compactness, the weak compactness and the complete continuity. Moreover, we gave the necessary and sufficient conditions for weighted composition operators to have closed range or to be Fredholm operators.[3] S.Ohno defined de Branges-Rovnyak spaces induced by composition operator and reduced elementary results from the general situation due to D.Sarason. We could consider the problems of multipliers and invariant subspaces.