TAILIEUCHUNG - Báo cáo toán học: "On some commutators of operators "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Trên một số commutators của các nhà khai thác. | J. OPERATOR THEORY 12 1984 47 - 64 Copyright by INCREST 1984À- ON SOME COMMUTATORS OF OPERATORS J. A ERDOS and s. G1OTOPOULOS The questions considered in this paper are of the following general conditions must operators Xand Ysatisfy so that their commutator XY YX belongs to a given set Such questions are essentially concerned with derivationsand have been extensively studied in relation to selfadjoint algebras. Non-selfadjoint theory has had less attention. However a significant part of the recent progress in nest algebra theory consists of solutions to problems of this type see 2 4 11 13 and 14 and it is in this context that the investigation is continued here. A set s of orthogonal projections on a Hilbert space H is called a nest if it is totally ordered. The set Alg ỹ of all bounded linear operators leaving invariant the range of each member of s is called the nest algebra of s. Nest algebras were introduced by Ringrose 18 and appear to be the most tractable-class of non-selfadjoint operator algebras. We use the following notation for any subalgebra sđ of - and any subset Xi of 77 which is a two-sided j -module C j . Xe H AX - XA e Ji for all A e st . In other words C jỉ Xi denotes the commutant of si modulo Xi. In this paper sd will always be either a nest algebra Alg ỹ its diagonal Q S or its core S and xt will always be some ideal of Alg f. It follows easily that C j Jt s AlgS in these cases. In 7 it is asked whether in general C AIg r Jt. This relation holds in all the cases considered here. Since every derivation of a nest algebra into J H is of the form A - AX XA for some X e J H 2 n the determination of C Alg ỹ Xi is equivalent to the determination of all derivations of Alg ỹ into XỈ. In Section 2 we use a method of Larson 16 to show that J Qi J when J is any of the diagonal ideals defined in 18 and 8 . From this we recover Larson s result 16 that C J Q J when J is the Jacobson radical of Alg f and show that the same relation holds when J is

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