Lecture notes on complex interpolation of compactness

TL;DR

This paper investigates whether compactness of linear operators is preserved under complex interpolation, providing new partial results especially when the target space is a UMD-space or via duality arguments.

Contribution

It offers new partial positive answers to the longstanding question about compactness preservation under complex interpolation, including cases involving UMD-spaces and duality.

Findings

Affirmative answer when target space is a UMD-space.

Positive result via duality for specific spaces.

Connection to earlier results and potential for a complete solution.

Abstract

Suppose that the linear operator $T$ maps $X_0$ compactly to $Y_0$ and also maps $X_1$ boundedly to $Y_1$. We deal once again with the 51 year old question of whether $T$ also always maps the complex interpolation space $[X_0,X_1]_\theta$ compactly to $[Y_0,Y_1]_\theta$. This is a short preliminary version of our promised technical sequel to our earlier paper arXiv:1410.4527 on this topic. It contains the following two small new partial results: (i) The answer to the above question is yes, in the particular case where $Y_0$ is a UMD-space. (ii) The answer to the above question is yes for given spaces $X_0$, $X_1$, $Y_0$ and $Y_1$ if the answer to the "dualized" or "adjoint" version of the question for the duals of these particular spaces is yes. In fact we deduce (i) from (ii) and from an earlier result obtained jointly by one of us with Nigel Kalton. It is remarked that a proof of a natural converse of (ii) would answer the general form of this question completely.