  In this paper it is proved that each densely defined $J$-skew-symmetric
operator (or each $J$-isometric operator with
$\overline{D(A)}=\overline{R(A)}=H$) in a Hilbert space $H$ has a
$J$-skew-self-adjoint (respectively $J$-unitary) extension in a Hilbert space
$\widetilde H\supseteq H$. We follow the ideas of Galindo in~[A.~Galindo, On
the existence of $J$-self-adjoint extensions of $J$-symmetric operators with
adjoint, Communications on pure and applied mathematics, Vol. XV, 423-425
(1962)] with necessary modifications.
