This is an announcement for the paper "Lecture notes on complex interpolation of compactness" by Michael Cwikel and Richard Rochberg.
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.
Archive classification: math.FA
Mathematics Subject Classification: Primary 46B70, 46B50. Secondary 46E15
Remarks: 7 pages
Submitted from: mcwikel@math.technion.ac.il
The paper may be downloaded from the archive by web browser from URL
http://front.math.ucdavis.edu/1411.0171
or