We assume that the conditions of question 4 are satisfied (i.e., $\lambda(i) = \lambda(0) - 2i$ for all $i \in \mathbf{Z}$). Let $\lambda, \mu \in \mathbf{C}^{\mathbf{Z}}$ and $F, H \in \mathcal{L}(V)$ defined by $H(v_i) = \lambda(i) v_i$ and $F(v_i) = \mu(i) v_{i+1}$. Show that $E \circ F = F \circ E + H$ if and only if for all $i \in \mathbf{Z}$,
$$\mu(i) = \mu(0) + i(\lambda(0) - 1) - i^2.$$