Complex interpolation of compact operators mapping into lattice couples
Abstract
Description
Suppose that (A_0,A_1) and (B_0,B_1) are Banach couples, and that T is a linear operator which maps A_0 compactly into B_0 and A_1 boundedly (or even compactly) into B_1. Does this imply that T maps [A_0,A_1]_s to [B_0,B_1]_s compactly for 0<s<1 ? (Here, as usual, [A_0,A_1]_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 (A_0,A_1) is arbitrary and (B_0,B_1) 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.
14 pages. (Page 13 contains routine and standard material which you quite probably will not need or want to print.) The only changes in this new version are the correction of small typographic errors on the first page and the updating of a reference.
14 pages. (Page 13 contains routine and standard material which you quite probably will not need or want to print.) The only changes in this new version are the correction of small typographic errors on the first page and the updating of a reference.