On a class of unitary operators on weighted Bergman spaces

In this paper we consider a class of weighted composition operators defined on the weighted Bergman spaces L2a (dA?) where D is the open unit disk in C and dA?(z) = (? + 1)(1 ? |z|2)?dA(z), ? > ?1 and dA(z) is the area measure on D. These operators are also self-adjoint and unitary. We establish here that a bounded linear operator S from L2a (dA?) into itself commutes with all the composition operators C(?) a , a ? D, if and only if B?S satisfies certain averaging condition. Here B?S denotes the generalized Berezin transform of the bounded linear operator S from L2a (dA?) into itself, C(?) a f = ( f ??a), f ? L2a (dA?) and ? ? Aut(D). Applications of the result are also discussed. Further, we have shown that ifMis a subspace of L?(D) and if for ? ? M, the Toeplitz operator T(?) ? represents a multiplication operator on a closed subspace S ? L2a (dA?), then ? is bounded analytic on D. Similarly if q ? L?(D) and Bn is a finite Blaschke product and M(?) q ( Range C(?) Bn) ? L2a (dA?), then q ? H?(D). Further, we have shown that if ? ? Aut(D), then N = {q ? L2a (dA?) : M(?) q (Range C(?)?) ? L2a (dA?)} = H?(D) if and only if ? is a finite Blaschke product. Here M(?)?, T(?)? , C(?)? denote the multiplication operator, the Toeplitz operator and the composition operator defined on L2a (dA?) with symbol ? respectively.