Complex interpolation of compact operators mapping into lattice couples
Abstract
Suppose that (A0,A1) and (B0,B1) are Banach couples, and that T is a linear operator which maps A0 compactly into B0 and A1 boundedly (or even compactly) into B1. Does this imply that T maps [A0,A1]s to [B0,B1]s compactly for 0<s<1 ? (Here, as usual, [A0,A1]s denotes the complex interpolation space of Alberto Calderon.) This question has been open for 44 years. Affirmative answers are known for it in many special cases. We answer it affirmatively in the case where (A0,A1) is arbitrary and (B0,B1) is a couple of Banach lattices having absolutely continuous norms or the Fatou property. Our result has some overlap with a recent result by Evgeniy Pustylnik.
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