Decompositions of Banach Lattices into Direct Sums

We consider the problem of decomposing a Banach lattice Z as a direct sum Z = X @ Y where X and Y are complemented subspaces satisfying a condition of incomparability (e.g. every operator from Y to X is strictly singular). We treat both the atomic and nonatomic cases. In particular we answer a question of Wojtaszczyk by showing that L1 fflL2 has unique structure as a nonatomic Banach lattice. One of the most important problems in the theory of Banach lattices, which is still open, is whether any complemented subspace of a Banach lattice must be linearly isomorphic to a Banach lattice. The main difficulty seems to lie in the fact that most of the criteria for a Banach space to be isomorphic to a lattice do not really distinguish between lattices and their complemented subspaces. We do not actually treat this question in the present paper but rather consider the situation Z = X d3 Y, where Z is a Banach lattice and X and Y two complemented subspaces which are assumed to satisfy different conditions that make them "distinct" in some or another sense. This line of research was initiated by P. Wojtaszczyk 128] (and also by I. S. Edelstein and P. Wojtaszczyk [3]) who proved that if Z has a normalized unconditional basis {Zn}n=l (i.e. it is a separable atomic lattice) so that every linear operator from Y into X is compact then {zn}n=l splits into two disjoint parts which are respectively equivalent to bases of X and Y. In particular, both X and Y have unconditional bases. The proof of this result is based on a fundamental theorem from [28 and 3], which is mentioned below as Theorem A. We give here a different proof which does not make use of Theorem A but instead is based on a simple "change of signs" result from [2], which is described below as Theorem B. We also consider the case when the compactness assumption above is replaced by the total incomparability of X and Y for which we prove a similar result provided X and Y have unconditional bases. Unfortunately, the most interesting case when every operator from Y into X is assumed to be strictly singular (which was raised as an open problem in [28]) remains unsolved. We conclude the section devoted to the atomic case with a simple theorem on block bases of a space with unconditional basis {zn}°°=1 whose span is complemented. Such a block basis splits into two disjoint parts, the first equivalent to a subsequence of {zn}n= Received by the editors September 5, 1986. 1980 Mathematics Subject Classification (1985 Revision). Primary 46B30, 46B15; Secondary 46A40. This research was supported by Grant No. 84-00210 from the United States-Israel Binational Science Foundation (BSF), Jerusalem, Israel. The research of the first author was also supported in part by NSF Grant DMS 8500938. The research of the second author was also supported in part by NSF Grant DMS 8601401. (r) 1987 American Mathematical Society OOO2-9947/87 $1.00 + $.25 per page