Past Functional Analysis Seminar

The Schur product is the commutative operation of entrywise

multiplication of two (possibly infinite) matrices. If we fix a matrix

A and require that the Schur product of A with the matrix of any

bounded operator is again the matrix of a bounded operator, then A is

said to be a Schur multiplier; Schur multiplication by A then turns

out to be a completely bounded map. The Schur multipliers were

characterised by Grothendieck in the 1950s. In a 2006 paper, Kissin

and Shulman study a noncommutative generalisation which they call

"operator multipliers", in which the theory of operator spaces plays

an important role. We will present joint work with Katja Juschenko,

Ivan Todorov and Ludmilla Turowska in which we determine the operator

multipliers which are completely compact (that is, they satisfy a

strengthening of the usual notion of compactness which is appropriate

for completely bounded maps).

A problem that has been open since (at least) 1960 is whether there exists an infinite-dimensional Banach space on which every bounded linear operator is a compact perturbation of a scalar multiple of the identity. The HI spaces constructed by Gowers and Maurey have "few operators" in a slightly weaker sense than this. Combining HI methods with a technique due to Bourgain, Spiros Argyros and the speaker have recently constructed a space which solves the original problem. The seminar talk will attempt to convey some of the underlying ideas.