Cyclic Vectors Of Analytic Function Spaces And Related Operator Theory

In this thesis we study characterizations of cyclic vectors in the logarithmicα-Bloch spaces,the logarithmic VMOA space,and the Q_p spaces in the unit disk.We give the characterizations of the zero-based invariant subspaces of the Q_p spaces.Prove the boundedness of the pointwise multiplication operator and the composition operator on the logarithmicα-Bloch spaces.Some necessary and sufficient conditions are given for which the weighted composition operator is bounded or compact on the the logarithmic Bloch space,between the logarithmic Bloch space and theα-Bloch spaces respectively.Charpter 1 gives the backgroud of the cyclic vectors of some analytic function spaces;the definitions of some analytic function spaces and the weighted composition operators are given;some known results which are close to the thesis are also listed.Charpter 2 concentrates in the logarithmicα-Bloch spaces in the unit disk. Some necessary and sufficient conditions are given for which the pointwise multiplication operator and the composition operator to be a bounded operator on the logarithmicα-Bloch spaces.A necessary and sufficient condition is given for which an analytic function is a cyclic vector for the little logarithmicα-Bloch spaces when 0＜α＜1,and some sufficient conditions and some necessary conditions are obtained for which an analytic function is cyclic for the little logarithmicα-Bloch spaces whenα≥1.Chapter 3 is devoted to investigate the boundedness and compactness of the weighted composition operator on the logarithmic Bloch space.Some sufficient and necesary conditions are given for weighted composition operators to be bounded or compact on the logarithmic Bloch space,between the logarithmic Bloch space and theα-Bloch spaces,between the little logarithmic Bloch space and theα-Bloch spaces,between the logarithmic Bloch space and the littleα-Bloch spaces respectively.Charpter 4 is concerns the logarithmic VMOA space in the unit disk.A sufficient condition and a necessary condition are obtained for which an analytic function is cyclic for the logarithmic VMOA space.Also,a sufficient condition and a necessary condition are given for which the pointwise multiplication operator is bounded on the logarithmic BMOA space and the logarithmic VMOA space.Charpter 5 aims at characterizations of cyclic vectors in the Q_p spaces and Q_p,O spaces in the unit disk.Some sufficient conditions and some necessary conditions for which an analytic function is a cyclic vector for the Q_p spaces and Q_p,O spaces.Meanwhile,some characterizations of the zero-based invariant subspaces of the Q_p spaces and Q_p,O spaces are given,and a necessary and sufficient condition is given for which the pointwise multiplication operator is a bounded operator in the Q_p,O spaces.