A general Extrapolation Theorem for absolutely summing operators

Abstract

In this note, we prove a general version of the Extrapolation Theorem for absolutely summing operators, extending the classical theorem due to B. Maurey ['Théorèmes de factorisation pour les opérateursà valeurs dans les espaces L p', Soc. Math. France, Asterisque 11, Paris, 1974]. Our result also contains the recent Extrapolation Theorem for Lipschitz p-summing operators as a particular case and also provides new extrapolation-type theorems.Theorem 1.1 (Extrapolation Theorem). Let 1 < r < p < ∞ and let X be a Banach space. If Π p (X; p ) = Π r (X; p ), then, for any Banach space Y,Besides its intrinsic elegance, the importance of the Extrapolation Theorem is easy to see; for instance, it can be used (see [10, p. 72]) to prove the famous Grothendieck theorem, which asserts that Π 1 ( 1 ; 2 ) = L( 1 ; 2 ).

Abstract extrapolation theorems for absolutely summing nonlinear operators

“…In Section 4 we complete the proof of the main result. In Section 5 we point out that our main result recovers the recent extrapolation theorem from [31] as a particular case and we show that this latter result holds with fewer assumptions.…”

“…In this section we show that the recent nonlinear extrapolation theorem proved in [31] is a particular case of our main result. Moreover, we show that one assumption of the extrapolation theorem of [31] can be dropped.…”

“…Moreover, we show that one assumption of the extrapolation theorem of [31] can be dropped. The main result of [31] is a generalization of the following: [25].) Let 1 < r < q < ∞ and X be a Banach space.…”

“…A natural question is whether extrapolation theorems hold for nonlinear summing operators. In this direction, several nonlinear extrapolation theorems have appeared recently (see, for example, [4,12,27,31]). In this paper we take advantage of the abstract framework constructed in [30] to prove an extrapolation theorem that encompasses all these previous results as particular cases.…”

“…The nonlinear theory of absolutely summing operators has been rapidly developed by several authors (see, e.g., [1,4,[6][7][8][10][11][12]18,20,[22][23][24][27][28][29][30][31]33]). A natural question is whether extrapolation theorems hold for nonlinear summing operators.…”