On the effect of rearrangement on complex interpolation for families of Banach spaces

TL;DR

This paper demonstrates that complex interpolation of Banach space families is generally unstable under rearrangement, except for specific transformations characterized as origin-preserving inner functions, linking the problem to matrix-valued Toeplitz operators.

Contribution

It provides a new proof of non-stability under rearrangement for families with more than two boundary values and characterizes invariant transformations using Toeplitz operator theory.

Findings

Rearrangement destabilizes complex interpolation for families with three or more boundary values.

Transformations invariant under complex interpolation are exactly the origin-preserving inner functions.

The method connects complex interpolation stability to matrix-valued Toeplitz operator theory.

Abstract

We give a new proof to show that the complex interpolation for families of Banach spaces is not stable under rearrangement of the given family on the boundary, although, by a result due to Coifman, Cwikel, Rochberg, Sagher and Weiss, it is stable when the latter family takes only 2 values. The non-stability for families taking 3 values was first obtained by Cwikel and Janson. Our method links this problem to the theory of matrix-valued Toeplitz operator and we are able to characterize all the transformations on $\T$ that are invariant for complex interpolation at 0, they are precisely the origin-preserving inner functions.