8
M. CWIKEL AND P. G. NILSSON
choices of weight functions WQ and w\ the couple of weighted lattices {XQ^WQ,X\,WI) is
a CM couple with constant independent of the choice of weights, then both Xo and
Xi must necessarily have the weak Fatou property.
Our central results establish two very different necessary and sufficient conditions
for any two couples X
w
= (XojWo,XijWl) and Y
v
= (YojVo,YijVl) of arbitrary saturated
a-order continuous weighted Banach lattices of measurable functions having the Fatou
property to be relatively CM for all choices of the weight functions wo,wi,vo and
v±. One of these conditions is a property which we call relative decomposability which
must be imposed on the lattices Xj and Yj for j = 0 and 1. It is defined by the
requirement that if / and g are any functions which can each be written as sums of
disjointly supported elements, / = Y^Li fn and g = Y£Li9n with \\gn\\Yj \\fn\\xj
for each n, and if / G Xj then g G Yj. (Cf. [CNS] and [CSc].) The second condition
tells us that to determine whether all relative interpolation spaces with respect to X
w
and Y
v
are relative K spaces for all choices of weights, it is in fact sufficient to check
this only for the special case of Calderon's complex interpolation spaces [Xw]9 and
[Yv]# for 0 6 1. Furthermore, if X = Y, then we only have to check for any two
arbitrarily chosen distinct values of 0. In that case our conditions are also equivalent
to the requirement that X must be a couple of (weighted) LP spaces. This shows
that Sparr's theorem [Sp, AC, Cwl] is the strongest possible positive result which is
invariant under change of weight functions for CM couples of Banach lattices.
Long ago, one of us asked whether, for an arbitrary compatible couple of Banach
spaces A, the condition that the complex interpolation spaces [A]# are all K spaces
might be sufficient to ensure that A is a CM couple. Although the results just men-
tioned would seem to reinforce the likelihood of an affirmative answer to this question,
it turns out, as shown in [MsO], that the answer in general is no. (Cf. also [Ms2].)
In the course of developing the machinery required here to prove our main theo-
rems, we obtain a number of auxiliary results which we consider to be also of indepen-
dent interest. Several of them will be described briefly in the next subsection.
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