Revista Matemática Iberoamericana
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Published online: 2015-07-16
On the effect of rearrangement on complex interpolation for families of Banach spacesYanqi Qiu (1) Aix-Marseille Université, Marseille, France
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 $\mathbb T$ that are invariant for complex interpolation at 0, they are precisely the origin-preserving inner functions.
Keywords: Complex interpolation method for families, rearrangement, matrix valued outer functions, Toeplitz operator, duality
Qiu Yanqi: On the effect of rearrangement on complex interpolation for families of Banach spaces. Rev. Mat. Iberoam. 31 (2015), 439-460. doi: 10.4171/RMI/840