  We present two alternative proofs of Mandrekar's theorem, which states that
an invariant subspaces of the Hardy space on the bidisc is of Beurling type
precisely when the shifts satisfy a doubly commuting condition. The first proof
uses properties of Toeplitz operators to derive a formula for the reproducing
kernel of certain shift invariant subspaces, which can then be used to
characterize them. The second proof relies on the reproducing property in order
to show that the reproducing kernel at the origin must generate the entire
shift invariant subspace.
