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

Let $\varphi$ 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 $1\le p<\mathrm{\infty }$ and $p\mathrm{\ne }2$. Our main result is the following trichotomy, which states that exactly one of the following alternatives holds: (i) $C_{\varphi }$ is a compact operator $H^p\to H^p$, (ii) $C_{\varphi }$ fixes a (linearly isomorphic) copy of ${\mathrm{\ell }}^p$ in $H^p$, but $C_{\varphi }$ does not fix any copies of ${\mathrm{\ell }}^2$ in $H^p$, (iii) $C_{\varphi }$ fixes a copy of ${\mathrm{\ell }}^2$ in $H^p$. Moreover, in case (iii) the operator $C_{\varphi }$ 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.