TAILIEUCHUNG - Báo cáo toán học: "Invariant operator ranges of nest algebras "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Phạm vi điều hành được sửa đổi của đại số tổ. | J. OPERATOR THEORY 3 1980 195 - 201 Copyright by INCREST 1980 INVARIANT OPERATOR RANGES OF NEST ALGEBRAS SING-CHEONG ONG Since the publication of the poineering paper 3 of Foias on invariant operator ranges of operator algebras much progress on this subject has been made by many others. Most of the existing literature on this subject is of the form assuming some conditions on the lattice of invariant operator ranges of operator algebras then something can be said about the algebras. In 8 Voiculescu calculates explicitly the lattices of invariant operator ranges of von Neumann algebras with Schwartz property p. Still very little is known about the lattice of invariant operator ranges of a given operator algebra. In this paper we determine the lattices of invariant operator ranges of some special nest algebras. 1. For a complex Hilbert space JF á denotes the algebra of bounded linear operators on ye. A submanifold Ổ ye is an operator range if there exists 7 0 cf. 2 in .ốỹ . F such that Tye Ì. For an algebra ja we denote the lattice cf. 2 of invariant operator ranges closed subspaces of every element of sđ by Lat1 2tsZ Lat . We adapt standard notation in 7 The following theorem of Foias 3 is of great importance. Theorem a. If sđ is a uniformly closed algebra and if Tyee Latj a z then there exists a unique bounded algebra homomorphism It of sđ into y ye such that AT Tn A and cdAi-ye N T 1 for all AS sđ. If contains a commutative self-adjoint subalgebra sđữ then T can be chosen positive and in the commutant of .y Combining the structure theorem of 5 and a moment s consideration of the null space we obtain the following useful theorem. Theorem B. Let p be a positive operator with p I and E the spectral measure of p. Ifan algebra sá leaves invariant the spectral subspaces E 2 k J j k 1 2 . then sể leaves the range pye invariant. 00 2. Let ye yej yPj a Hilbert space of dimension J be an orthogonal j i decomposition of ye. Let sđ be the algebra of operators on ye leaving .

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