On non-linear mappings preserving the semi-inner product

TL;DR

This paper investigates the property (SL) in smooth normed spaces, characterizing when semi-inner product preserving maps are linear, and constructs examples of spaces with or without this property.

Contribution

It provides new characterizations of property (SL), examples of spaces lacking it, and constructs a space where all isomorphic smooth spaces have property (SL).

Findings

Spaces isomorphic to ll_p have property (SL) for 1 < p < inite.

A smooth, strictly convex Banach space without property (SL) is constructed.

A space is constructed where all isomorphic smooth spaces possess property (SL).

Abstract

We say that a smooth normed space $X$ has a property (SL), if every mapping $f:X \to X$ preserving the semi-inner product on $X$ is linear. It is well known that every Hilbert space has the property (SL) and the same is true for every finite-dimensional smooth normed space. In this paper, we establish several new results concerning the property (SL). We give a simple example of a smooth and strictly convex Banach space which is isomorphic to the space $\ell_p$, but without the property (SL). Moreover, we provide a characterization of the property (SL) in the class of reflexive smooth Banach spaces in terms of subspaces of quotient spaces. As a consequence, we prove that the space $\ell_p$ have the property (SL) for every $1 < p < \infty$. Finally, using a variant of the Gowers-Maurey space, we construct an infinite-dimensional uniformly smooth Banach space $X$ such that every smooth Banach space isomorphic to $X$ has the property (SL).