We study composition operators acting between WEIGHTED BERGMAN SPACES with admissible Bekolle weights. The boundedness and compactness of composition operators are characterized in terms of the generalized Nevanlinna counting function associated with the inducing map of the composition operator and the associated weight function of BERGMAN space. For a special case, we also give the estimate of the essential norm.

For an analytic self-map φ of the unit disk D in the complex plane C, a nonnegative integer n, and u analytic function on D, WEIGHTED di erentiation composition operator is de ned by (D n φ ; u f)(z) = (n) u(z)f (φ (z)), where f is an analytic function on D and z 2 D. In this paper, we study the boundedness and compactness of D n φ ; u, from WEIGHTED BERGMAN SPACES with admissible weights to Bloch-type SPACES.

Let j,y be the analytic self-maps of the unit ball B, we characterize the Hilbert-Schmidt differences of two composition operator Cj and Cy on WEIGHTED BERGMAN space A2a, and give some conclusions about the topological structure of C(A2a), the space of all bounded composition operators on A2a endowed with operator norm.

We obtain some new sharp results for some new analytic functional SPACES defined with the help of BERGMAN metric ball.

‎In this paper‎, ‎we first study the boundedness of generalized‎ WEIGHTED composition operators from minimal Möbius invariant‎ ‎subspace to the $n$th WEIGHTED type space‎. ‎Then‎, ‎we obtain‎ ‎equivalence conditions for its boundedness by using the Bell‎ ‎polynomials‎. ‎We also find some estimates for the essential‎ ‎norm of this operator and use them to present equivalence‎ ‎conditions for the compactness of such an operator‎.