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Differentiation in star-invariant subspaces. II: Schatten class criteria. (English) Zbl 1011.47006
Given an inner function $$\theta$$ on the upper half-plane, let the space $$K_\theta: =H^2\cap \theta\overline {H^2}$$ be such that the operator $${d\over dy}: K_\theta\to L^2$$ is compact. As follows from the results obtained in first part of the paper [ibid. 364-386 (2002; Zbl 1011.47005)], then $$\theta$$ is necessarily a Blaschke product $$B$$ whose zeros $$z_j$$ satisfy $$\text{Im} z_j\to\infty$$, and the operator $${d\over dx}$$ is henceforth regarded as acting from $$K_B$$ to $$K_{B^2}$$. The problem under consideration is to determine when the operator belongs to the Schatten-von Neumann classes $$s_p$$ (i.e., if its $$s$$-numbers belong to $$l^p)$$. A complete solution is given for $$p=1$$ and $$p=2$$ (the cases of nuclear and Hilbert-Schmidt operators), and explicit formulas for the trace and the Hilbert-Schmidt norm are presented. For other values of $$p$$, some necessary and some sufficient conditions are found.

##### MSC:
 47A15 Invariant subspaces of linear operators 30D50 Blaschke products, etc. (MSC2000) 30D55 $$H^p$$-classes (MSC2000) 47B10 Linear operators belonging to operator ideals (nuclear, $$p$$-summing, in the Schatten-von Neumann classes, etc.)
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