Cyclic Invariant Subspaces

If M is an invariant subspace of H2 (G), the proof of Corollary 6.1.6. shows that $$ \mathcal{N}: = C_{\alpha ^{ - 1} } \circ \mathcal{M}$$ is a nearly invariant subspace of . We know from Corollary 3.2.9 that if {0, ∞} is not a subset of the common zeros of N and Φ and Ψ are the normalized reproducing kernels at z=0 and z=∞, then the smallest nearly invariant subspace containing Φ and Ψ is equal to N. From Remark 6.2.12 we also see that Φ ○ α is the normalized reproducing kernel for M at α1(0) while Ψ ○ α is the normalized reproducing kernel at α−1(∞).