2008/02/24 by Michael Ćwikel, Michael Cwikel, Cwikel, Michael · 1 citation
Mathematics · #46B70 #46E30 (Primary) #Advanced Banach Space Theory #Advanced Harmonic Analysis Research #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory #math.FA #msc:46B70 #msc:46E30
paper · pdf · doi:10.48550/arxiv.0802.3520
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.
openalex publication_date 2008/02/24 · arxiv created 2008/06/30 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.