Complex interpolation of compact operators mapping into the couple (FL^{\infty},FL_{1}^{\infty})

Abstract

If (A_0,A_1) and (B_0,B_1) are Banach couples and a linear operator T from A_0 + A_1 to B_0 + B_1 maps A_0 compactly into B_0 and maps A_1 boundedly into B_1, does T necessarily also map [A_0,A_1]_s compactly into [B_0,B_1]_s for s in (0,1)? After 42 years this question is still not answered, not even in the case where T is also compact from A_1 to B_1. But affirmative answers are known for many special choices of (A_0,A_1) and (B_0,B_1). Furthermore it is known that it would suffice to resolve this question in the special case where (B_0,B_1) is the special couple (l^\infty(FL^\infty), l^\infty(FL^\infty_1)). Here FL^\infty is the space of all sequences which are Fourier coefficients of bounded functions, FL^\infty_1 is the weighted space of all sequences (a_n) such that (e^n a_n) is in FL^\infty, and thus B_0 and B_1 are the spaces of bounded sequences of elements in these spaces (i.e., they are spaces of doubly indexed sequences). We provide an affirmative answer to this question in the related but simpler case where (B_0,B_1) is the special couple (FL^\infty,FL^\infty_1).