Closed range composition operators on Hilbert function spaces

We show that a Carleson measure satisfies the reverse Carleson condition if and only if its Berezin symbol is bounded below on the unit disk DD. We provide new necessary and sufficient conditions for the composition operator to have closed range on the Bergman space. The pull-back measure of area measure on DD plays an important role. We also give a new proof in the case of the Hardy space and conjecture a condition in the case of the Dirichlet space.