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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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