Differentiation in star-invariant subspaces II. Schatten class criteria

Given an inner function 0 on the upper half-plane C+, let K-0 ;= H-2 circle minus thetaH(2) be the corresponding star-invariant subspace of the Hardy space H-2. Earlier we showed that the differentiation operator d/dx : K-0 --> L-2(R) is bounded iff theta' is an element of L-infinity(R) and compact iff theta' is an element of C-0(R). The current problem is to determine when the above operator belongs to the Schatten-von Neumann class f(p). The most important cases are p = 1 and p = 2, and for these p's we solve the problem completely. The f(1) and f(2) criteria that arise involve the decay rate of theta' at infinity or (alternatively) the distribution of the zero sequence {z(j)} of 0. Moreover, explicit formulae for the trace and the Hilbert-Schmidt noun are provided. For other values of p, we point out some necessary and some sufficient conditions in order that d/dx is an element of f(p). The gap is presumably quite small, and we are able to eliminate it for special classes of zero sequences {z(j)}. (C) 2002 Elsevier Science (USA).