On Quotients of Banach Spaces Having Shrinking Unconditional Bases

It is proved that if a Banach space Y is a quotient of a Banach space having a shrinking unconditional basis, then every normalized weakly null sequence in Y has an unconditional subsequence. The proof yields the corollary that every quotient of Schreier's space is c o-saturated. §0. Introduction. We shall say that a Banach space Y has property (WU) if every normalized weakly null sequence in Y has an unconditional subsequence. The well known example of Maurey and Rosenthal [MR] shows that not every Banach space has property (WU) (see also [O]). W.B. Johnson [J] proved that if Y is a quotient of a Banach space X having a shrinking unconditional f.d.d. and the quotient map does not fix a copy of c 0 , then Y has (WU). Our main result extends this (and solves Problem IV.1 of [J]). Theorem A. Let X be a Banach space having a shrinking unconditional finite dimensional decomposition. Then every quotient of X has property (WU). Of course such an X will itself have property (WU). Furthermore, if (E n) is an unconditional f.d.d. (finite dimensional decomposition) for X, then (E n) is shrinking if and only if X does not contain ℓ 1. The proof of Theorem A yields Theorem B. Let Y be a Banach space which is a quotient of S, the Schreier space. Then Y is c o-saturated. Y is said to be c o-saturated if every infinite dimensional subspace of Y contains an isomorph of c 0. Our notation is standard as may be found in the books of Lindenstrauss and Tzafriri [LT 1,2]. The proof of Theorem A is given in §1 and the proof of Theorem B appears in §2. §3 contains some open problems. We thank H. Rosenthal and T. Schlumprecht for useful conversations regarding the results contained herein. §1. The proof of Theorem A.