Let $\lambda, \mu \in \mathbf{C}^{\mathbf{Z}}$ and $H \in \mathcal{L}(V)$ defined by $H(v_i) = \lambda(i) v_i$. Show that $H \circ E = q^2 E \circ H$ if and only if for all $i \in \mathbf{Z}, \lambda(i) = \lambda(0) q^{-2i}$.
Let $\lambda, \mu \in \mathbf{C}^{\mathbf{Z}}$ and $H \in \mathcal{L}(V)$ defined by $H(v_i) = \lambda(i) v_i$. Show that $H \circ E = q^2 E \circ H$ if and only if for all $i \in \mathbf{Z}, \lambda(i) = \lambda(0) q^{-2i}$.