Differentiation in Star-Invariant Subspaces II. Schatten Class Criteria

✍ Scribed by Konstantin M. Dyakonov

• Publisher: Elsevier Science
• Year: 2002
• Tongue: English
• Weight: 201 KB
• Volume: 192
• Category: Article
• ISSN: 0022-1236
• DOI: 10.1006/jfan.2001.3921

✦ Synopsis

Given an inner function y on the upper half-plane C þ , let K y :¼ H 2 É yH 2 be the corresponding star-invariant subspace of the Hardy space H 2 . Earlier we showed that the differentiation operator d dx : K y ! L 2 ðRÞ is bounded iff y 0 2 L 1 ðRÞ and compact iff y 0 2 C 0 ðRÞ. The current problem is to determine when the above operator belongs to the Schatten-von Neumann class S p . The most important cases are p ¼ 1 and p ¼ 2, and for these p's we solve the problem completely. The S 1 and S 2 criteria that arise involve the decay rate of y 0 at infinity or (alternatively) the distribution of the zero sequence fz j g of y. 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 d dx 2 S p . The gap is presumably quite small, and we are able to eliminate it for special classes of zero sequences fz j g.