"Commutators and the structure of isomorphisms on L_p"
Abstract:In a recent joint work with Dosev and Johnson we characterized the (bounded, linear operators which are) commutators on L_p spaces, i.e. the operators on L_p which are of the form AB-BA, as exactly those operators which cannot be written as aI+S with non zero a, and S which is "L_p strictly singular" namely not an isomorphism when restricted to any subspace isomorphic to L_p. The proof combines tools previously used in characterization of commutators on simpler spaces together with new results on the structure of isomorphisms on L_p spaces.
The talk will concentrate on this later part which strengthen results
from a work of Johnson, Maurey, Tzafriri and the speaker. We show in
particular that any subspace of L_p, 1 < p < \infinity , which is
isomorphic to L_p contains a further subspace isomorphic to
and on which there is a bounded linear projection from L_p; the
constants involved do not depend on the original isomorphism but only
on p. Although the proof is quite involved I would like to try and
explain some of its ideas, most of which go back to the work with
Johnson, Maurey and Tzafriri.