Zero subspaces of polynomials on l1(Gamma)

TL;DR

This paper constructs specific quadratic polynomials on l_1 spaces that exhibit unique properties regarding their zero subspaces, including examples with both separable and nonseparable maximal zero subspaces.

Contribution

It introduces two new examples of quadratic polynomials on l_1 spaces with distinct zero subspace structures, highlighting novel behaviors.

Findings

First polynomial has both separable and nonseparable maximal zero subspaces.

Second polynomial's zero subspaces are all separable despite uncountable index set.

Provides insights into the structure of zero subspaces in polynomial mappings.

Abstract

We provide two examples of complex homogeneous quadratic polynomials P on Banach spaces of the form l_1(I). The first polynomial P has both separable and nonseparable maximal zero subspaces. The second polynomial P has the property that while the index-set I is not countable, all zero subspaces of P are separable.