The Properties Of Some Operators On Analytic Function Spaces

The operator theory in analytic function space is one of the focuses of functional theory,since 1960 s,Nordgren has given sufficient conditions for composite operators to be bounded operators in Hardy spaces,operator theory has attracted many scholars at home and abroad,and obtained a lot of profound theoretical achievements,gradually formed a complete system.This paper is devoted to studying the related properties of RiemannStieltjes type operator from QK(p,q)space to Zygmund space in unit disk and the related properties of weigthed composition operator on the Bloch space with logarithmic weight in unit polydisc.This article consists of five chapters as follows:In the first chapter,we summarize historical background,development course,research purposes and research meaning of analytic function space and operator theory.And we make a simple introduction of the operator theory on QK(p,q)space and Bloch type space and existing results.In the second chapter,we introduction the basic knowledge such as QK(p,q)space,Bloch type space,Riemann-Stieltjes type operator and weighted composition operator,and some marks and lemmas.In the third chapter,we discuss Riemann-Stieltjes operator Jg,φand Ig,φfrom QK(p,q)space to Zygmund space,by choosing testing function of QK(p,q)space and using operator theory and the properties of analytic function,the related theorem of the two operators Ig,φ: QK(p,q)→ Z(Z0)and Jg,φ: QK(p,q)→ Z(Z0)are obtained.In the fourth chapter,firstly,we discuss the related properties of the weighted composition operator on the Bloch space with logarithmic weight in the case of several complex variables in unit polydisc in Cn,we extend the case of one complex variable in unit disk considered,and then,we discuss the related properties of the weighted composition operator on the small Bloch space with logarithmic weight,and we also extend some existing results on weighted small Bloch space in the unit polydisc.In the fifth chapter,we summarize the conclusions obtained in this paper and put forward some questions for further studies in this research field.