  In this paper, we shall characterize the components of the polar
decomposition for an arbitrary $J$-unitary operator in a Hilbert space. This
characterization has a quite different structure as that for complex symmetric
and complex skew-symmetric operators. It is also shown that for a $J$-imaginary
closed symmetric operator in a Hilbert space there exists a $J$-imaginary
self-adjoint extension in a possibly larger Hilbert space (a linear operator
$A$ in a Hilbert space $H$ is said to be $J$-imaginary if $f\in D(A)$ implies
$Jf\in D(A)$ and $AJf = -JAf$, where $J$ is a conjugation on $H$).
