### Complex interpolation of compact operators

Welcome to this little corner of cyberspace, which is dedicated to encouraging everyone and anyone to solve or help solve a more than half-century old problem about interpolation of compact operators by Alberto Calderon’s complex method.

I will be very happy and grateful to receive your comments and suggestions at:

mcwikel@math.technion.ac.il

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 Calderon’s celebrated paper about complex interpolation.

SO THIS PROBLEM IS NOW FIFTY-SEVEN YEARS OLD!!!

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.

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16.08.2015 Some new special cases in which there is a positive answer to Calderon’s problem are discussed in s joint paper http://arxiv.org/abs/1411.0171 with Richard Rochberg.

It can be considered as a sequel to another joint paper http://arxiv.org/abs/1410.4527 with Richard Rochberg, which gives a relatively recent survey of this problem

I am currently writing another paper which will identify some additional special cases for which there is a positive answers.

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.

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Here is a much older document (meanwhile revised), 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.)

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An earlier paper with Nigel Kalton is available here.