Radial Operators And The Berezin Transform On The Bergman Space Of The Unit Ball

In this paper,we investigate the connection between compactness of operators on the Bergman space and the boundary behaviour of the corresponding Berezin transform. We prove that for a class of operators that we call radial operators,an oscillation criterion and diagonal are sufficient conditions under which the compactness of an operator is equivalent to the vanishing of the Berezin transform on the unit sphere.We further study a special class of radial operators,that is,Toeplitz operators with a radial L~1(B_n) symbol.The paper is mainly divided into four chapters.The first section is the introduction of the whole paper.We talk about the background of this paper,and make plans for the research of the problem.The next section consists of definitions and fundamental theorems.We state a definition and some lemmas on radial bounded operators for the proof of main results.The third section we present a necessary and sufficient for the radial bounded operator A to be compact on the Bergman space in the unit ball in terms of the Berezin transform approach for A,and also determine a family of operators which satisfy the conclusion.At last,we summarize the results of the whole paper.