I suggest that the “birthday” of this problem could be taken to be August 15, 1963, the day on which the editors of Studia Mathematica received Alberto's celebrated paper about complex interpolation.
Good luck to us all in this little enterprise.
Michael Cwikel
I plan to insert new material at this place on the page, if and when it becomes available. So you will find the earlier announcements pushed further down on this page.
27.10.08 Here is a new joint paper with Eliahu Levy which is not exactly about the main problem of this page but is connected to my earlier paper about compact operators mapping into couples of lattices.
14.3.08.
The pdf file at
http://www.maths.sussex.ac.uk/Seminars/document/seminar-slides-07-007.pdf
is from a lecture reporting on
work by Sören Bartels, Max Jensen and Rüdiger Müller on a finite element method for the miscible
displacement problem.
On pages 24-29 of this pdf file you can see complex interpolation of compact operators being applied in this
work. The preprint of the paper about this research is available at
http://www.mathematik.hu-berlin.de/publ/pre/2008/P-08-02.pdf . The authors found it convenient to use a result in
my paper with Nigel Kalton (available below). However, much as I would like to claim that our result is the only way
to meet
their needs, I should mention that there is also another way, a bit longer and less convenient for non-interpolators:
One can check that the "range" couple in this context satisfies the
special "approximation" property used by Calderon in his original paper
about these things. So Calderon's theorem also applies here.
Another special case of Calderon's problem has now been resolved. This is case where the range couple is a couple of Banach lattices of measurable functions on the same underlying measure space, satisfying some mild conditions. For details see this.
24.6.06. You can now get Svante's and my new preprint, which gives a positive answer to THE question in the case where the "range" couple (B_0,B_1) is (FL^\infty,FL^\infty_1). You can get it from the ArXiv or from Svante's web page.
2.3.06 There is now a short paper by Svante Janson and me which, apart from briefly mentioning Fedja's very interesting example (see below) also briefly mentions some other new developments. You can get this paper from Svante's web page or click directly here for the PDF version or PS version.
Some particularly interesting observations about this problem have
recently been
made by Fedor Nazarov. You can see the details
here.
Fedja's example in the above document provides a negative answer to a
question which I asked at a meeting in Oberwolfach in August 2004.
The
summary of my
(now somewhat outdated)
Oberwolfach talk is on
pages 2110 to 2113 of
this document,
together with summaries of some other talks on
compactness and interpolation at that meeting.
.
Here is a much older document,
written as an informal supplement to the “toolbox” paper about this
problem which I wrote jointly with Natan Krugljak and Mieczsylaw
Mastylo. (Illinois J. Math. 40 (1996) 353-364.)
.
An earlier paper with Nigel Kalton is available
here.