Complex interpolation of compact operators mapping into lattice couples

TL;DR

This paper proves that under certain conditions, complex interpolation preserves the compactness of linear operators between Banach couples, resolving a 44-year-old open problem.

Contribution

It establishes that if (A_0,A_1) is arbitrary and (B_0,B_1) are Banach lattices with specific properties, then compactness is preserved under complex interpolation.

Findings

Affirmative answer for Banach lattices with absolutely continuous norms or Fatou property.

Extends known results to arbitrary (A_0,A_1) couples.

Addresses a 44-year-old open question in interpolation theory.

Abstract

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.