TAILIEUCHUNG - Báo cáo toán học: "A generalization of Koosis-Lax interior compactness theorem "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Một khái quát Koosis-Lax lý chặt nội thất. | J. OPERATOR THEORY 8 1982 197 - 208 Copyright by INCREST 1982 A GENERALIZATION OF KOOSIS-LAX INTERIOR COMPACTNESS THEOREM s. V. HRUẫÓEV and A. L. VOLBERG 1. INTRODUCTION It follows from von Neumann spectral multiplicity theorem that given an increasing continuous chain GJ 5 0 of closed subspaces of a Hilbert space G. there exists a direct integral of Hilbert spaces Too H ị H dm t 0 and a unitary operator S -. G - H such that PG l 0 3 t Here stands for the projection of H onto the direct integral ị Hsdm 0 and m denotes a positive measure. In the present paper we treat a special family of chains in the Hardy class H2. Namely let 0 be an inner function corresponding to a positive singular Borel measure J. on the unit circle T C 6 c ICl 1 ớ z exp I ị Ị- í W T and let Gt - Kyt H2 WH2 Í 0. It is clear that the family ơt I 0 forms a continuous chain in the following sense 1 Gs G s C t 2 Go - 0 and _J Gt is dense in H2 t 0 3 for each s 0 u Gt is dense in Gs and f l Gt Gs. t s t s 198 s. V. HRUSCEV and A. L. The explicit formulae for the direct integral H and for IF proved in 7 give immediately that oo H - - dr I _ 1 and that the values of the operator on the rational fractions k z 1 1- -lz 1 which span H- are given by 2 0 z Í -Xz -1. We shall study the interior compactness property for the chains Egt l 9. To be more precise we introduce a semigroup of translation operators Tj ỈĨ - li C - . 4- h C h 0 for fe H. Every open interval J of the half-line R 0. oo detines a semi-norm Ịị J r j I. Ơ5 1 J A subspace E in H is called translation invariant if zbE c E for every h h 0. We call such a space E interior compact abbreviated TUC-space if the set e E ij 4 1 is precompact with respect to the semi-norm 4 whenever closZ J. This definition is due to p. D. Lax 8 . ĩn 9 p. D. Lax proved the following remarkable theorem. Theorem. P. D. Lax Let E be a translation invariant subspace of IL Then E is interior compact iff zh E - E is a compact operator for all h. Now the problem

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