On the stability of the differential process generated by complex interpolation

TL;DR

This paper investigates the stability properties of the differential process in complex interpolation of Banach spaces, revealing conditions for global bounded stability and limitations when increasing the number of spaces involved.

Contribution

It extends previous work by establishing stability results for families of up to three K"othe spaces and identifying instability for larger families, also providing local and isometric stability results.

Findings

Global bounded stability for up to three K"othe spaces on arcs.

No bounded stability for four or more K"othe spaces.

Local and global isometric stability for pairs of Banach spaces.

Abstract

We study the stability of the differential process of Rochberg and Weiss associated to an analytic family of Banach spaces obtained using the complex interpolation method for families. In the context of K\"othe function spaces we complete earlier results of Kalton (who showed that there is global bounded stability for pairs of K\"othe spaces) by showing that there is global (bounded) stability for families of up to three K\"othe spaces distributed in arcs on the unit sphere while there is no (bounded) stability for families of four or more K\"othe spaces. In the context or arbitrary pairs of Banach spaces we present local stability results and global isometric stability results.