Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Về các tính chất trừu tượng của lý trí (1-điểm) approximants. | J. OPERATOR THEORY 6 1981 217 -231 Copyright by INCREST 198Ĩ WEAK COMPACTNESS IN BANACH LATTICES CONSTANTIN p. NICULESCU INTRODUCTION In his remarkable paper devoted to the study of the weakly compact operators on C S -spaces Grothendieck 5 has proved that a bounded subset K of C S is relatively weakly compact if and only if lim sup Ấ ơ Ị 0 for each sequence of n pairwise disjoint open subsets 0 of s. This result was used later by H. p. Rosenthal 16 to formulate the notion of a relatively disjoint family of measures which is. instrumental in his criterion of weak compactness in a space A bounded subset K of z j p which is not weakly compact contains a basic sequence .Y which is equivalent to the natural basis of zx and such that Span .r is complemented in . The main result of our first section extends this criterion for weakly sequentially complete Banach lattices. Alternatively we obtain the following dichotomy theorem for operators T given on Banach spaces and taking values in weakly sequentially complete Banach lattices T is either weakly compact or its restriction to a complemented subspace which is isomorphic to is an isomorphism. Section 2 is devoted to characterizing the behaviour of weakly compact operators defined on Banach lattices in terms of unconditional basic sequences. For example it is proved that every operator Tdefined on a Banach lattice which contains no lattice isomorph of Zj is weakly compact provided that T maps weak Cauchy sequences into norm convergent sequences the reciprocal Dunford-Pettis property . A special attention is paid to finding the conditions under which the following dichotomy result a variation of the Dieudonné property holds Each operator T from E to a Banach space Fis either weakly compact or the restriction of T to a subspace isomorphic to -0 is an isomorphism. In the present paper we shall often make use of the classical results due to Kakutani concerning the AM and AL-spaces. Particularly we shall need the .
