For α > −1, let Aα denote the corresponding weighted Bergman space of the unit ball. For any self-adjoint subset G ⊂ L∞, let T(G) denote the C∗−subalgebra of B(Aα) generated by {Tf : f ∈ G}. Let CT(G) denote the commutator ideal of T(G). It was showed by D. Suárez (in 2004 for n = 1) and by the author (in 2006 for all n ≥ 1) that CT(L∞) = T(L∞) in the case α = 0. In this paper we show that in t...

Composition operators between weighted Bergman spaces with a smaller exponent in the target space are studied. An integrability condition on a generalized Nevanlinna counting function of the inducing map is shown to characterize both compactness and boundedness of such an operator. Composition operators mapping into the Hardy spaces are included by making particular choices for the weights.

We compare the rate of decay singular numbers a given composition operator acting on various Hilbert spaces analytic functions unit disk D. show that for Hardy and Bergman spaces, our results are sharp. also give lower upper estimates with symbol “cusp map” lens maps, weighted Dirichlet spaces.