  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.
