Let $\lambda, \mu \in \mathbf{C}^{\mathbf{Z}}$. We define the linear maps $F, H \in \mathcal{L}(V)$ respectively by
$$H(v_i) = \lambda(i) v_i \quad \text{and} \quad F(v_i) = \mu(i) v_{i+1}, \quad i \in \mathbf{Z}.$$
Show that $H \circ E = E \circ H + 2E$ if and only if for all $i \in \mathbf{Z}, \lambda(i) = \lambda(0) - 2i$.