# Jerusalem Mathematics Colloquium

The Jerusalem Mathematics Colloquium this week is dedicated to the memory of
Prof. Lior Tzafriri z"l

Thursday, 4th March 2010, 4:00 pm

Mathematics Building, Lecture Hall 2

##

Gideon Schechtman

(Weizmann)

"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
*L_p*
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.

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