9 - Mixed summing norms and finite dimensional Lorentz spaces

Summary

INTRODUCTION

This largely expository article describes some recent results concerning the mixed summing norm Π_{p, 1} as applied to operators on ℓ_∞-spaces (especially the finite-dimensional spaces). The whole subject of summing operators and their norms can be said to have started with a result on Π_{2, 1} – the theorem of Orlicz (1933) that the identity operators in ℓ₂ and ℓ₁ are (2, 1)-summing. However, since that time the study of mixed summing norms has been somewhat neglected in favour of the elegant and powerful theory of the “unmixed” summing norms Πl_p. A breakthrough hasnow been provided by the theorem of Pisier, which does for mixed summing norms what the fundamental theorem of Pietsch does for unmixed ones (see e.g.). One version of Pisier's theorem states that the operator can be factorised through a Lorentz function space L_{p, 1} (λ). Such spaces were introduced in, and are discussed in and. However, it is not easy to find a really simple outline of the definition and basic properties of these spaces adapted to the (obviously simpler) finite-dimensional case, so the present paper includes a brief attempt to provide one.

It is of particular interest to compare the value of Π_{2, 1} and Π₂ for operators on ℓ_∞ or. It is a well-known fact, underlying the famous Grothendieck inequality, that there is a constant K, independent of n, such that for all operators T from to ℓ₁ or ℓ₂, we have Π₂ (T)≤ K∥T∥. This equates to saying that Π₂ (T) s K'Π_{2, 1} (T) for such T.