Studies of The Spaces of Analytic Functions and Their Operators

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(1) In 1995, Ohno investigated Toeplitz and Hankel operators on harmonic Borgman spaces on the unit disk. Main results are to characterize algebraic properties, boundedness and compactness. There exists a relation between the compactness of Hankel operators and Bourgain algebras. This is a very interesting problem. In 1996, he studied the conditions that differences of two composition operators are compact. He obtained some examples and a necessary condition closely related to the compactness of one composition operator.(2) Funabashi studied 5-dimensional submanifolds of a nearly Kaehler 6-spherc in the purcly imaginary octonians. Main result is that for any hypersurface of 6-sphere, there exists a grobal quaternion structure on the contact distribution. Moreover he studied tublar hypersurfaces. He iedentified the symplectic group SP (1) with the 3-dimensional sphere and considered parametrized 3-dimcnsional submanifolds in terms of SP (1) -orbits in the 6-sphere.(3) Hashimoto investig … More ated submanifolds theory in a 6-dimensional sphere S^6. A 6-dimensional sphere has an almost Hermitian structure.It was proved that n-dimensional sphere admit almost complex structures except for n*2,6. Also the automorphism group of this almost Hcrmitian structure of S^6 coincide with the exceptional Lie group G_2. The 2-dimensional submanifolds of a 6-dimensional sphere is called the J-holomorphic curves of S^6 if its tangent space is invariant under the almost complex structure. I obtained some classification theorems and a rigidity theorem with respect to the Lie group G_2 about J-holomorphic curves of S^6.(4) Ishizaki has studied the complex differential equations, mainly admissible solutions of first order algebraic differential equations and complex oscillation for an equation of the form f"+A (z) f=0. Complex dynamics theory has been also of our great interest. Study of hypertranscendency has treated from the two points of view, say complex differential theory and complex dynamics theory. Less