TAILIEUCHUNG - Báo cáo toán học: "Operators of type A and local absolute continuity "

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Các nhà khai thác của loại A và địa phương liên tục tuyệt đối. | J. OPERATOR THEORY 13 1985 49-61 Copyright .by INCREST 1985 OPERATORS OF TYPE A AND LOCAL ABSOLUTE CONTINUITY CONSTANTIN p. NICULESCU Let be a Banach lattice F a Banach space and T G L E F . T is said to be of type A provided that 0 xn ị in implies 7a- is norm convergent. This class of operators was first considered by Dodds 6 who noted the connection with the class of all weakly compact operators defined on C S spaces. The terminology is motivated by the fact that the identity of a ơ-complete Banach lattice is of type A if and only if is of type A in the Kantorovich s terminology . has order continuous norm. The aim of the present paper is to develop a parallel to the measure theory and to prove the analogue of the following result due to Bartie Dunford and Schwartz Given a finite additive measure m on a ơ-algebra ST m is ơ-additive if and only if m is absolutely continuous with respect to a suitable ơ-additive measure Ậi ST - R . The concept of absolute continuity was extended in 12 to operators given on Banach lattices. Let and be as above and let x G x 0. We shall say that an operator T G is locally absolutely continuous with respect to x . T loc x provided that for every 0 and every X G X ỷ 0 there exists a 5 0 such that y G X and x l l 5 implies T Our main result asserts that if has a quasi-interior point then an operator T G is of type A if and only if T is locally absolutely continuous with respect to a suitable x G x 0 see Theorem below. This result includes the Bartle-Dunford-Schwartz theorem mentioned above and is strongly related to a result of Amemya which states that all Hausdorff order continuous locally solid topologies on a vector lattice induce the same topology on an order bounded set . 4--2495 50 CONSTANTIN p. NICULESCU As a consequence we are able to prove that the operators of type A admit nice factorizations through order continuous Banach lattices see Theorem below. The latter fact is used to derive operator theoretical .

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