  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.
