# Some new partial answers to a 52 year old interpolation question

It is now more than 52 years since Studia Mathematica received Alberto

Calder\'on's very remarkable paper about his theory of complex

interpolation spaces. And one of the questions which Calder\'on

implicitly asked in that paper, by solving it in a significant special

case, is apparently still open today:

DOES COMPLEX INTERPOLATION PRESERVE THE COMPACTNESS OF AN OPERATOR?

After briefly surveying attempts to solve this question over several

decades, I will also report on a few new partial answers obtained

recently, some of them (arXiv:1411.0171) jointly with Richard

Rochberg. Among other things there is an interplay with Jaak Peetre's

"plus-minus" interpolation method, (arXiv:1502.00986) a method which

probably deserves to be better known. Banach lattices and UMD spaces

also have some roles to play.

Several distinguished mathematicians have expressed the belief that

that the general answer to this question will ultimately turn out to be

negative. Among other things, I will try to hint at where a counterexample

might perhaps be hiding. You are all warmly invited to seek it out,

or prove that it does not exist.

A fairly recent survey which discusses this question is available at

arXiv:1410.4527.

- Last modified: 5/11/2015