Nearly invariant subspaces

Abstract

For a general domain , a subspace¹ N ⊂ H² (Ω) (not necessarily invariant) is said to be nearly invariant if there is some λ₀ ∈ Ω such that whenever f ∈ N and f (λ₀) = 0, then

$$ \frac{f} {{z - \lambda _0 }} \in N. $$

The following useful proposition, found in [7, Prop. 5.1], says that we need not single out a particular λ₀.

Recall from the introduction that a subspace will be a closed linear manifold.