Some Extensions of Witt's Theorem

TL;DR

This paper generalizes Witt's theorem to various types of simultaneous isometries of subspaces, providing necessary and sufficient conditions for extending isometries under different structural constraints.

Contribution

It introduces new conditions for extending isometries to larger spaces while preserving complex structures like flags and decompositions, broadening the theorem's applicability.

Findings

Extended Witt's theorem to simultaneous isometries

Derived necessary and sufficient conditions for isometry extensions

Applied results to generic flags and subspace pairs

Abstract

We extend Witt's theorem to several kinds of simultaneous isometries of subspaces. We determine sufficient and necessary conditions for the extension of an isometry of subspaces $\phi:E\to E'$ to an isometry $\phi_V:V\to V'$ that also sends a given subspace to another, or a given self-dual flag to another, or a Witt's decomposition to another and a special self-dual flag to another. We also determine sufficient and necessary conditions for the isometry of generic flags or the simultaneous isometry of (subspace, self-dual flag) pairs.