Generalized inverses and polar decomposition of unbounded regular operators on Hilbert $C^*$-modules

TL;DR

This paper characterizes when unbounded regular operators on Hilbert $C^*$-modules have polar decompositions, linking it to the existence of generalized inverses and properties of the underlying $C^*$-algebra.

Contribution

It establishes necessary and sufficient conditions for polar decomposition of unbounded regular operators on Hilbert $C^*$-modules, connecting it to the structure of the $C^*$-algebra.

Findings

Operators have polar decomposition iff their ranges are orthogonally complemented.

Existence of generalized inverses for operators is equivalent to the algebra being of compact operators.

Characterization of when all densely defined operators have polar decomposition or generalized inverses.

Abstract

In this note we show that an unbounded regular operator $t$ on Hilbert $C^*$-modules over an arbitrary $C^*$ algebra $ \mathcal{A}$ has polar decomposition if and only if the closures of the ranges of $t$ and $|t|$ are orthogonally complemented, if and only if the operators $t$ and $t^*$ have unbounded regular generalized inverses. For a given $C^*$-algebra $ \mathcal{A}$ any densely defined $\mathcal A$-linear closed operator $t$ between Hilbert $C^*$-modules has polar decomposition, if and only if any densely defined $\mathcal A$-linear closed operator $t$ between Hilbert $C^*$-modules has generalized inverse, if and only if $\mathcal A$ is a $C^*$-algebra of compact operators.