Composition Operators Belonging to Operator Ideals

With each composition operator C fsf(w acting between classical Bergman w spaces Ap and Aq, 1Fp, q-`, we associate in a natural way a sequence of non-negative scalars j k , kgN. Our main result states that the composition operator belongs to a prescribed quasi-Banach ideal A if and only if the diagonal operator D : lpªlqgiven by a .¬ a j . belongs to A. Conversely, we show j k k k that for any l )max 2rqy2rp, 0.and k gR there exists a composition operator C : ApªAqwhose associated sequence is equivalent to.kyl log k.k, kG2. w This provides an effective tool to separate quasi-Banach ideals by means of composition operators. The results can be generalized to standard weighted Bergman spaces and beyond.