On The Weak Compactness of Composition Operators

• Main Author: 陳明玉
• Other Authors: Lin, Ing-Jer
• Format: Others
• Language: en_US
• Online Access: http://ndltd.ncl.edu.tw/handle/68779567997774080482

碩士 === 國立高雄師範大學 === 數學研究所 === 86 === In this thesis, we are wainly concerned with the weak compactness of composition operator on the Hardy Space H1 of the bidisc. The main result of thf thesis is : Theorem : Let Cφbe a composition operator on H1(△2). Then the following statements are equivalent: (i) Cφ: H1(△2)→H1(△2) is compact; (ii) Cψ: H1(△2)→H1(△2) is weakly compact; (iii) C*ψ: BMO→ VMO is bounded. The subject of the thesis is motivated by Sarason's result on the equivalence of weak and norm compactness of composition operator on the Hardy Space H1 of the open unit disc △. In 1994, Lang Chen proved that it holds on the Bergman space of strongly pseudoconvex domains. In this thesis, I considered the composition operator on the Hardy space H1 of the bidisc. This theorem is proved in the Hardy space H1 of the bidisc. The proof requires duality theorems for the BMO and VMO spaces, in which case it is based on some fundamental work of Ing-Jer Lin, on Factorization and Duality in the Hardy spaces of the polydisk.