denote the spaces of analytic functions in the unit disk such that
for a normal weight function
, respectively. The author completely characterizes the boundedness and compactness of the composition between differentiation and multiplication operator
and also for the operators
is a given analytic function and
is a non-constant analytic self-map on the disk, and between the spaces
. Due to the fact that
as particular cases, his results allow him to unify a number of previously known theorems.