It is known that for many, but not all, compatible couples of Banach spaces
(AQ, A\) it is possible to characterize all interpolation spaces with respect to the couple
via a simple monotonicity condition in terms of the Peetre if-functional. Such couples
may be termed Calderon-Mityagin couples. The main results of the present paper pro-
vide necessary and sufficient conditions on a couple of Banach lattices of measurable
which ensure that, for all weight functions
and w\, the couple
of weighted lattices
is a Calderon-Mityagin couple. Similarly, necessary
and sufficient conditions are given for two couples of Banach lattices
(Yo,Yi) to have the property that, for all choices of weight functions wo,wi,Vo and
vi, all relative interpolation spaces with respect to the weighted couples (XoiWo,XiiWl)
and (Yb^cp^i^i) m a y ^ e described via an obvious analogue of the above-mentioned
if-functional monotonicity condition.
A number of auxiliary results developed in the course of this work can also be
expected to be useful in other contexts. These include a formula for the if-functional
for an arbitrary couple of lattices which offers some of the features of Holmstedt's
formula for K(£, / ; Lp, L9), and also the following uniqueness theorem for Calderon's
spaces XQ~ Xf: Suppose that the lattices Xo, X±, YQ and Y\ are all saturated and
have the Fatou property. If X^~eXf = Y^~eYf for two distinct values of 0 in (0,1),
then Xo = YQ and X\ = Y\. Yet another such auxiliary result is a generalized version of
Lozanovskii's formula (XQ~ Xf) = (-^o) ~ C^i) f°r ^ n e associate space
1991 Mathematics Subject Classifications: 46B70, 46E30.
Key words and phrases: Banach lattice of measurable functions, Kothe func-
tion space, weighted lattice, interpolation space, if-space, Calderon couple, Calderon-
Mityagin couple, relatively decomposable spaces.
Received by the editor February 14, 2000, and in revised form September 20, 2002.
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