Quasisimilarity of invariant subspaces for $C_0$ operators with multiplicity two

For an operator $T$ of class $C_0$ with multiplicity two, we show that the quasisimilarity class of an invariant subspace $M$ is determined by the quasisimilarity classes of the restriction $T\mathrm{\mid }M$ and of the compression $T_{M^{\perp }}$. We also provide a canonical form for the subspace $M$.