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

Abstract

In this article, we prove the existence of the polar decomposition of 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 U0 such that T=U0T. In fact U0 is unique if N(U0) = N(T). In particular, if H is separable and U is a partial isometry with T=UT, then we prove that U = U0 if and only if either N(T) = {0} or R(T)⊥ = {0}.