Extension of vector-valued integral polynomials

Daniel Carando,Silvia Lassalle

doi:10.1016/j.jmaa.2004.10.020

Abstract

We study the extendibility of integral vector-valued polynomials on Banach spaces. We prove that an X-valued Pietsch-integral polynomial on E extends to an X-valued Pietsch-integral polynomial on any space F containing E, with the same integral norm. This is not the case for Grothendieck-integral polynomials: they do not always extend to X-valued Grothendieck-integral polynomials. However, they are extendible to X-valued polynomials. The Aron–Berner extension of an integral polynomial is also studied. A canonical integral representation is given for domains not containing ℓ 1 .

Highlights

In this note we study extendibility properties of Pietsch and Grothendieck integral polynomials
Since we focus on Grothendieck and Pietsch integral polynomials, we discuss the preservation of the respective integral norms
We prove that a Pietsch-integral polynomial P : E → X extends to an X-valued Pietsch-integral polynomial over any F ⊃ E, with the same integral norm

Summary

Introduction

In this note we study extendibility properties of Pietsch and Grothendieck integral polynomials. It is important to remark that in the definition, the extension of P must be X-valued Another consideration to take into account regarding extendibility is the preservation of the norm. The third section deals with the Aron–Berner extension of a (Pietsch or Grothendieck) integral polynomial. We show that this extension is integral, with the same integral norm. We refer to [14,22] for notation and results regarding polynomials in general, to [13, 16,23,24] for tensor products of Banach spaces and to [1,2,13,15] for integral operators, polynomials and multilinear mappings

Definitions and general results

Extension of integral polynomials

The Aron–Berner extension of an integral polynomial