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Theorem 1.  Given a number c >= 1, there exists a J-unitary operator \hat{V}, such that:  (a) r(\hat{V})= r(\hat{V}^{-1})= c ;  (b) S(c^{-1}\hat{V})=S(c^{-1}\hat{V}^{-1})  =S(c^{-1}\hat{V}^{*-1}) = S(c^{-1}\hat{V}^*)={0}  (c) there exist maximal strictly positive and strictly negative  \hat{V}^{\pm 1}-{invariant subspaces}  L_{+}, L_{-}, such that they are mutually J-orthogonal and  L_{+} + L_{-} is dense in the space.  (d_1) if L_1 is non-zero \hat{V}-invariant subspace, thenr(\hat{V}|L_1)=r(\hat{V})  (d_2) if L_2 is non-zero \hat{V}^{-1}-invariant subspace, thenr(\hat{V}^{-1}|L_2)=r(\hat{V}^{-1}) .

An Example of J-unitary Operator. Solving a Problem Stated by M.G. Krein