Isometries of James-Schreier and Lorentz spaces

TL;DR

This paper characterizes the isometries of James-Schreier and Lorentz sequence spaces, showing that the only isometries on the James-Schreier space are the identity and its negative, and establishing a Banach-Stone-type result for Lorentz spaces.

Contribution

It provides a complete description of the isometry group of the James-Schreier space and extends Banach-Stone-type theorems to Lorentz sequence spaces.

Findings

Only ±Id are isometries on V_1.

Characterization of extreme points of the dual unit ball.

Banach-Stone-type theorem for Lorentz sequence spaces.

Abstract

We study the group of surjective linear isometries on certain real Banach sequence spaces using the preservation of extreme points in the closed unit ball. Our main result provides a characterization of the extreme points of the dual unit ball of the James-Schreier space $V_1$. As a consequence, we show that the only isometries on $V_1$ are $\pm Id$. We also obtain a Banach-Stone-type result for Lorentz sequence spaces, analogous to one proved in \cite{CarothersIsometrisLorentz} for Lorentz function spaces.