  Given a lineal H_0 and x_0\in H_0 and a linear injective operator U_0: H_0
\to H_0 such that all U_0^N, N \in {\bf Z} exist and all U_0^N x_0, N \in {\bf
Z} are linearly independent, anyone can define on span{{U_0}^N x_0 | N \in {\bf
Z}} a (pre)hilbert scalar product such that U_0 becomes a unitary operator.
  The problem under consideration is: Suppose there is specified an indefinite
inner product {,}_0 on H_0 and U_0 is a {,}_0-unitary operator.
  Can one introduce a (pre)hilbert topology on L_{x_0} so that after completion
and possible extension the resulting {,}_ext is continuous, the resulting U_ext
is {,}_ext-unitary and there exists a pair L_{+}, L_{-} mutually
{,}_ext-orthogonal, maximal strictly positive and respectively negative
subspaces, so that they are U_ext-invariant? More generally, can one construct
a sequence (chain) of transformations of the type "restrict, change topology,
make completion, extend, restrict,..." with the same result? (And, as a result,
after some transformations, which are natural in the field of indefinite inner
product spaces, U_ext will become usual Hilbert space unitary operator).
  For a relatively wide class of pairs "operator, inner product" positive
solutions proposed.
