Complex Interpolation of Compact Operators

If (A0, A1) and (B0, B1) are Banach couples and a linear operator T : A0 +A1 → B0 +B1 maps A0 compactly into B0 and maps A1 boundedly into B1, does T necessarily also map [A0, A1]θ compactly into [B0, B1]θ for θ ∈ (0, 1)? After 42 years this question is still not answered, not even in the case where T : A1 → B1 is also compact. But affirmative answers are known for many special choices of (A0, A1) and (B0, B1). Furthermore it is known that it would suffice to resolve this question in the special case where (B0, B1) is the special couple (`∞(FL∞), `(FL1 )). Here FL ∞ is the space of all sequences {λn}n∈Z which are Fourier coefficients of essentially bounded functions, and FL1 is the weighted space of all sequences {λn}n∈Z such that {eλn}n∈Z ∈ FL∞. We provide an affirmative answer to this question in the related but simpler case where (B0, B1) is the special couple (FL∞, FL1 ).