New Banach space properties of the disc algebra and H∞

Abstract

J. BOURGAINwhere/~ is a probability measure, I the identity map andj the canonical embedding.The space of strictly p-integral operators is denoted by Ip(X, Y) and is equipped with the (strictly integral) norm i (u) = inf[ISI111711where the infimum is taken over all factorizations.Say that X has Grothendieck property provided any operator from X into ! 2 is 1-summing, thus B(X,/2)=III(X, /2).An equivalent formulation is the equality B(X*, I~)=II2(X*, 1~).Grothendieck's theorem asserts that L~(a)-spaces have Grothendieck property.As pointed out in [44] (Theorem 3.2), this general result follows easily from the fact that the operator --~l, where P is a Paley projection, is onto.This shows the usefulness of certain specific operators arising in harmonic analysis to the general theory.It is shown in [41] (Theorem 94) that Grothendieck's theorem can be improved to the equality B(l 1,/2)=ii0(ll,/2).A way of seeing this (cf.[32], section 2) is to consider the set A=Z+U {-2n; n=0,1,2 .... } and the orthogonal projection Q:CA___~ 2 L{-En} which is again onto by Paley's theorem.Now, for p>O, one has the inequality \ ,,2 f' an[ 2) <~if~f~J ~ In=~ane-t~~ lPdO}l/P \ from which it follows that Q is p-summing.Absolutely summing operators on A appear in the study of certain multipliers.For instance, Paley's theorem that each (A, P)-multiplier M is/2-summable is equivalent to the statement MEH2(A, lJ).In this spirit, the reader is referred to [35] for a study of translation-invariant absolutely summing operators.Our work actually shows that these results extend to arbitrary operators and that the equality B(A, /1)=1-I2(A, /l) holds in general.One of the striking facts about operators on the disc algebra is the following extension of the coincidence of the notions of p-summing and p-integral operators on C(K)-spaces (see [44], section 2).

Locations

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