Research on families of functions determining structures of spaces of analytic functions

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Izuchi (Head investigator) introduced the concept of weak infinite powers of Blaschke products and characterized Blaschke products which are weak generators of L^∞. Also Izuchi defined L^1 and L^∞-type singular inner functions and showed that for a positive singular measure there exists a particular set in M (H^∞) associated with the measure. He proved the existence of trivial points which are not contained in the closure of a nontrivial Gleason part. This answers the Budde problem. And the set of such points is dense in the set of trivial points.Izuchi showed the existence of homeomorphic parts which are not locally sparse. This answers Gorkin-Mortini problem. He determined closed ideals of H^∞ whose zero sets are contained in the set of nontrivial points (with Gorkin-Mortini). Also Izuchi solved the Gorkin-Mortini problem concerning with prime ideals in H^∞+C.About invariant subspaces on T^2, Izuchi studied A_φ-invariant subspaces and gen-eralized Nakazi's theorems (with Matsugu). And he started to study composition operators on H^∞ and determined connected components with respect to the essential norm topology (with Zheng).On the results of investigators, Saito determined invariant subspaces on T^2 under the certain condition. Huruya defined the concept of p-hyponormality for n-tuple operators, and proved Putnam type inequality. Hayashi showed that Myrberg phenomenon follows from the uniquness theorem under some additinal conditions. Hatori solved Lausen-Neumann's problem concerned with commutative Banach al-gebras. On a Bohr group, Tanaka proved that an invariant subspace generated by a single function if and only if its cocycle is cohomologous to a singular cocycle. Takagi studied multiplicative and composition operators on ^p-spaces.