  We study idempotent, model, and Toeplitz operators that attain the norm.
Notably, we prove that if $\mathcal{Q}$ is a backward shift invariant subspace
of the Hardy space $H^2(\mathbb{D})$, then the model operator $S_{\mathcal{Q}}$
attains its norm. Here $S_{\mathcal{Q}} = P_{\mathcal{Q}}M_z|_{\mathcal{Q}}$,
the compression of the shift $M_z$ on the Hardy space $H^2(\mathbb{D})$ to
$\mathcal{Q}$.
