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It is shown that if  T  is a      C ⋅  0 {C_{ \cdot 0}}       contraction with finite defect indices, then Hyperlat  T  is (lattice) generated by those subspaces which are either      ker ⁡  ψ  ( T ) \ker \psi (T)       or ran      ran ξ  ( T ) ¯  \overline {{\text {ran}}\;\xi (T)}      , where      ψ  \psi       and      ξ  \xi       are scalar-valued inner functions.

The hyperinvariant subspace lattice of a contraction of class 𝐶.₀