Let $B, C$ be two $2\times 2$ matrices with complex entries. Let an infinite block matrix $H$ be given by its component: $H_{n,n'} = B \delta_{n,n'+1} + B^\dagger \delta_{n,n'-1} + C \delta_{n,n'} $. Here, $(n,n')\in\mathbb{Z}^2$.
How can I find the inverse of $H$?
Note: I came to this problem by trying to find the Green's function of a nearest-neighbor Hamiltonian.