On the polar decomposition of right linear operators in quaternionic Hilbert spaces

TL;DR

This paper establishes the existence and uniqueness conditions of the polar decomposition for densely defined closed right linear operators in quaternionic Hilbert spaces, extending classical results to the quaternionic setting.

Contribution

It proves the existence of the polar decomposition for such operators and characterizes the uniqueness of the partial isometry involved.

Findings

Existence of polar decomposition for densely defined closed right linear operators in quaternionic Hilbert spaces.

Uniqueness of the partial isometry when the null spaces coincide.

Conditions under which the partial isometry in the decomposition is unique.

Abstract

In this article we prove the existence of the polar decomposition for densely defined closed right linear operators in quaternionic Hilbert spaces: If $T$ is a densely defined closed right linear operator in a quaternionic Hilbert space $H$, then there exists a partial isometry $U_{0}$ such that $T = U_{0}|T|$. In fact $U_{0}$ is unique if $N(U_{0}) = N(T)$. In particular, if $H$ is separable and $U$ is a partial isometry with $T = U|T|$, then we prove that $U = U_{0}$ if and only if either $N(T) = \{0\}$ or $R(T)^{\bot} = \{0\}$.