On the structure of invariant subspaces for isometric composition operators on $H^{2}(\mathbb{D})$ and $H^{2}(\mathbb{C}_{+})$

Abstract

We consider the isometric composition operators $C_g$ on $\cal H$, where $\cal H$ is the Hardy space $H^2$ on the open unit disc $\mathbb{D}$ or the right half plane $\mathbb{C}_{+}$, defined by $C_g(f) = f \circ g$. It follows from Nordgren’s results that in the case ${\cal H} = H^{2}(\mathbb{D})$, $C_g$ is isometric if and only if g is an inner function such that g(0)=0. In the case ${\cal H} = H^{2}(\mathbb{C}_{+})$ we characterize the analytic functions g on $\mathbb{C}_{+}$ such that $C_g$ is isometric. Then we give explicit constructions of invariant subspaces $\cal M$ for $C_g$ such that ${C_g}_{|{\cal M}}$ is similar to an infinite direct sum of unilateral shifts. It follows that the lattice of invariant subspaces of $C_g$ is extremely rich. Finally we characterize the common invariant subspaces of $C_g$ and the unilateral shift or the shift semigroup corresponding to nonsingular inner functions.