Differentiation in Star-Invariant Subspaces II. Schatten Class Criteria

Given an inner function θ on the upper half-plane C+, let Kθ≔H2⊖θH2 be the corresponding star-invariant subspace of the Hardy space H2. Earlier we showed that the differentiation operator ddx: Kθ→L2(R) is bounded iff θ′∈L∞(R) and compact iff θ′∈C0(R). The current problem is to determine when the above operator belongs to the Schatten–von Neumann class Sp. The most important cases are p=1 and p=2, and for these pʹs we solve the problem completely. The S1 and S2 criteria that arise involve the decay rate of θ′ at infinity or (alternatively) the distribution of the zero sequence {zj} of θ. Moreover, explicit formulae for the trace and the Hilbert–Schmidt norm are provided. For other values of p, we point out some necessary and some sufficient conditions in order that ddx∈Sp. The gap is presumably quite small, and we are able to eliminate it for special classes of zero sequences {zj}.