The Berezin Transform And Radial Operators On Weighted Bergman Space

The Berezin transform of the operator is closely related with its compactness. People search for the necessary and sufficient conditions of compact operator by researching the corre-sponding Berezin transform of the operator on different spaces. Using the Berezin transform of the operator, one can translate the knowledge of operator theory into the describing of properties of functions. In1998, Axler S and Zheng D C have proved that finite sums of finite products of Toeplitz operators, which the vanishing of the Berezin transform on the unit disk implies that the operator is compact. In2002, under certain conditions, for some operators, Zorboska have discussed the vanishing of its Berezin transform on the boundary of the unit disk D implies that the operator is compact. This paper is to study the Berezin transform of the operator and the compactness of radial operator on the weighted Bergman space Aα2（D）. Under certain condi-tions, we demonstrate a class of radial operators, which the vanishing of the Berezin transform on the boundary of D indicates that the operator is compact, and research some properties of the essential commutant of Tz.In chapter1, we study the Berezin transform of the operator on the weighted Bergman space Aα2（D）. By two examples, it is shown that the vanishing of the Berezin transform on the boundary of D implies that the operator is not compact.In chapter2, we discuss the radial operator on weighted Bergman space Aα2（D） and intro-duce the radialization of the Berezin transform of the operator and the Berezin transform of the radial operator.In chapter3, for a class of the radial operator on the weighted Bergman space, we find a necessary and sufficient condition of the compact operator A which vanishing of the Berezin transform on the boundary of D.In chapter4, we discuss some properties of the essential commutant of Tz on the weighted Bergman space Aα2（D）.The main results are:Proposition3.2.1Let A be a bounded radial operator on the weighted Bergman space Aα2（D） with diagonal {an}, such that n （an-an-1） is bounded. Then A（z）â†'0,as|z|â†'1-implies that A is compact.Proposition3.3.2Let∫be a bounded radial function in D. Then the following three conditions are equivalent: （ⅰ）Tf:Aa2（D）â†'Aa2（D） is compact;（ⅱ）f（z）â†'0as|z|â†'1-；（ⅲ）1/(（1-x）^1+a)fx1f（（?）t）（1-t）αdtâ†'0as xâ†'1-...