Hyperbolic Q <sub> T S </ Sub> On The Space And The Analytic Function Space Operator Theory,

P.Wu studied the criterion of normal and Bloch functions with a nondecreasing function in the paper On increasing functions,Bloch functions and normal functions in 1998, on the basic of this paper, P.Wu and H.Wulan formally introduce the definition of QK spaces in their paper Characterizations of QK spaces. By now QK spaces also attracts a lot of attentions of many people. At the same time, there is a lot of unsolved problems in this field. Recently, P.Wu and K.Zhu made a lot of research in the properties of Composition Operators between the QK spaces and other spaces. Such as, P.Wu and H.Wulan gave the characterizations of bounded and compact composition operators from Bloch spaces to QK spaces, X.Cheng in the paper Composition operator between the weighted Bloch space and QK space studied the boundedness of the composition operators between weighted bloch space and QK space. J.Zhou in the paper About the analytic QT,s spaces in 2009 formally introduce the definition of QT,s spaces and the inclusion relations in this space. In recent years, Space of hyperbolic functions has become a research focus, X.Li introduce the definition of hyberbolic Qp spaces in his master's thesis in 2003, and investigate its primary properties.This thesis mainly consite of four parts:Chapter 1 and chapter 2 introduce the background of the Function spaces and operator theory and some main research results.Chapter 3 we have the hyperbolic QT,s space QT,s* by combining the hyperbolic derivative and the classic QT,s space, investigate some basic properties of hyberbolic QT,s space, include use Carleson measure to charactering hyberbolic QT,s space.Chapter 4 mainly discuss the necessary and sufficient condition when the composition operators Cφfrom Blog (B0,log) space to QT,s space are bounded and compact.Chapter 5 mainly discuss the necessary and sufficient conditions when the Volterra Operator from Bloch type analytic function spaces Bαto weighted bloch type analytic function spaces Blogβ are bounded and compact.