Rigidity of composition operators on the Hardy space $H^p$

Let $\phi$ be an analytic map taking the unit disk $\mathbb{D}$ into itself. We establish that the class of composition operators $f \mapsto C_\phi(f) = f \circ \phi$ exhibits a rather strong rigidity of non-compact behaviour on the Hardy space $H^p$, for $1\le p < \infty$ and $p \neq 2$. Our main result is the following trichotomy, which states that exactly one of the following alternatives holds: (i) $C_\phi$ is a compact operator $H^p \to H^p$, (ii) $C_\phi$ fixes a (linearly isomorphic) copy of $\ell^p$ in $H^p$, but $C_\phi$ does not fix any copies of $\ell^2$ in $H^p$, (iii) $C_\phi$ fixes a copy of $\ell^2$ in $H^p$. Moreover, in case (iii) the operator $C_\phi$ actually fixes a copy of $L^p(0,1)$ in $H^p$ provided $p > 1$. We reinterpret these results in terms of norm-closed ideals of the bounded linear operators on $H^p$, which contain the compact operators $\mathcal K(H^p)$. In particular, the class of composition operators on $H^p$ does not reflect the quite complicated lattice structure of such ideals.