Stable invariant subspaces for operators on Hilbert space

If T is a bounded operator on a separable complex Hilbert space ℋ, an invariant subspace ℳ for T is stable provided that whenever ${\displaystyle \left\{T_n\right\}}$ is a sequence of operators such that ${\displaystyle \Vert T_n-T\Vert \to 0}$, there is a sequence of subspaces ${\displaystyle \left\{{ℳ}_n\right\}}$, with ${\displaystyle {ℳ}_n}$ in ${\displaystyle Lat{T}_n}$ for all n, such that ${\displaystyle P_{{ℳ}_n}\to P_{ℳ}}$ in the strong operator topology. If the projections converge in norm, ℳ is called a norm stable invariant subspace. This paper characterizes the stable invariant subspaces of the unilateral shift of finite multiplicity and normal operators. It also shows that in these cases the stable invariant subspaces are the strong closure of the norm stable invariant subspaces.