The author considers the classes of

$J$-symmetric operators and

$J$-selfadjoint operators on a Hilbert space with respect to an antilinear involution

$J$, as well as various related classes. These classes should not be confused with the similar classes of operators on a Krein or Pontryagin space. Some specific features of matrix representations of

$J$-symmetric and

$J$-skew-symmetric operators are studied. The main result of the paper provides conditions under which a bounded linear operator can be represented as a product of a

$J$-unitary operator and a

$J$-selfadjount one. A good bibliography concerning operators on spaces with an antilinear involution is given.