Isometries of Grassmann spaces, II

TL;DR

This paper characterizes surjective isometries of the Grassmann space of all infinite rank and corank projections on a separable Hilbert space, using geodesic analysis, extending previous finite-rank results.

Contribution

It provides the first complete description of surjective isometries on the set of all infinite rank and corank projections, employing a novel geodesic-based approach.

Findings

Complete characterization of isometries on infinite rank and corank projections

New proof technique based on geodesic analysis

Alternative proof for finite rank projections

Abstract

Botelho, Jamison, and Moln\'ar \cite{BJM}, and Geh\' er and \v{S}emrl \cite{GeS} have recently described the general form of surjective isometries of Grassmann spaces of all projections of a fixed finite rank on a Hilbert space $H$. As a straightforward consequence one can characterize surjective isometries of Grassmann spaces of projections of a fixed finite corank. In this paper we solve the remaining structural problem for surjective isometries on the set $P_\infty (H)$ of all projections of infinite rank and infinite corank when $H$ is separable. The proof technique is entirely different from the previous ones and is based on the study of geodesics in the Grassmannian $P_\infty (H)$. However, the same method gives an alternative proof in the case of finite rank projections.