Generalized inverses and polar decomposition of unbounded regular operators on Hilbert $C^*$-modules
Michael Frank, Kamran Sharifi
TL;DR
The note proves that an unbounded regular operator on a Hilbert $C^{*}$-module has a polar decomposition if and only if the closures of its range and the range of $|t|$ are orthogonally complemented, which is also equivalent to the existence of unbounded regular generalized inverses for $t$ and $t^*$. It also shows that for any densely defined $\mathcal{A}$-linear closed operator $t$, polar decomposition and generalized inverses exist precisely when the coefficient algebra $\mathcal{A}$ is a $C^{*}$-algebra of compact operators. The results connect polar decompositions, generalized inverses, and orthogonal decompositions with the structure of $\mathcal{A}$, extending Hilbert-space facts to Hilbert $C^{*}$-modules and clarifying when bounded transforms and projections arise.
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.