Isometries between spaces of homogeneous polynomials

Christopher Boyd,Silvia Lassalle

doi:10.1016/j.jfa.2005.01.009

Abstract

We derive Banach–Stone theorems for spaces of homogeneous polynomials. We show that every isometric isomorphism between the spaces of homogeneous approximable polynomials on real Banach spaces E and F is induced by an isometric isomorphism of E ′ onto F ′ . With an additional geometric condition we obtain the analogous result in the complex case. Isometries between spaces of homogeneous integral polynomials and between the spaces of all n -homogeneous polynomials are also investigated.

Highlights

Isometries between Banach spaces are those morphisms which preserve the metric structure of the spaces
In 1932 Banach [3] showed that if K and L are compact metric spaces and T is an isometric isomorphism from C(K), the space of continuous real valued functions on K, to C(L), the space of real valued functions on L, there is a homeomorphism from L into K and a continuous function h on L with |h(y)| = 1 such that (Tf )(y) = h(y)f ◦ (y) for all f in C(K) and all y in L
Since Banach’s and Stone’s results, it has emerged that isometries between a wide range of Banach function spaces have the above form with the condition on h relaxed to be an element of the range rather than satisfy |h(y)| = 1

Summary

Introduction

Isometries between Banach spaces are those morphisms which preserve the metric structure of the spaces. The space of all nuclear n-homogeneous polynomials on E is denoted by PN(nE) and becomes a Banach space when the norm of P is given as the infimum of. We define the integral norm of an integral polynomial P , P I, as the infimum of taken over all regular Borel measures which satisfy (1) It is shown in [20] (see [21, Section 2.2]) that PI(nE ) is isometrically isomorphic to PA(nE) via the Borel transform B given by B In addition it is shown that this result is true for the classes of nuclear, approximable, K-bounded, integral, extendible nhomogeneous polynomials along with the space of n-homogeneous polynomials which are weakly continuous on bounded sets irrespective of further conditions on E or F. To [23] for further information on isometries of Banach spaces

Canonical and power-preserving mappings

Isometries between spaces of approximable polynomials on real Banach spaces

Isometries of spaces of approximable polynomials on complex Banach spaces

Isometries between other spaces of homogeneous polynomials