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

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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.