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

Izuchi（首席研究员）引入了Blaschke乘积的弱无穷幂的概念，并刻画了作为L^∞的弱生成元的Blaschke乘积。此外，Izuchi定义了L^1和L^∞型奇异内函数，并证明了对于正奇异测度，M（H^∞）中存在与该测度相关联的特定集合。他证明了存在的平凡点不包含在封闭的一个非平凡的格里森部分。这就回答了巴德问题。Izuchi证明了非局部稀疏的同胚部分的存在性。这就回答了Gorkin-Mortini问题。他确定了H^∞的闭理想，其零点集包含在非平凡点的集合中（与Gorkin-Mortini一起）。Izuchi还解决了H^∞+C中与素理想有关的Gorkin-Mortini问题。关于T^2上的不变子空间，Izuchi研究了A_φ-不变子空间，推广了Nakazi定理（与Matsugu）。他开始研究H^∞上的复合算子，并确定了关于本质范数拓扑的连通分支（与郑先生）。在研究者的结果基础上，Saito在一定条件下确定了T^2上的不变子空间。Huruya定义了n元组算子的p-亚正规性的概念，并证明了Putnam型不等式。Hayashi证明了Myrberg现象在某些附加条件下是由唯一性定理得出的。Hatori解决了与交换Banach代数有关的Lausen-Neumann问题。在Bohr群上，Tanaka证明了由单个函数生成的不变子空间当且仅当其上循环与奇异上循环上同调。高木研究了^p-空间上的乘法和复合算子。

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.