A Wold-type decomposition for commuting isometric pairs

We obtain a Wold-type decomposition theorem for an arbitrary pair of commuting isometries $V$ on a Hilbert space. More precisely, $V$ can be uniquely decomposed into the orthogonal sum between a bi-unitary, a shift-unitary, a unitary-shift and a weak bi-shift part, that is, a part $S=(S_1,S_2)$ that can be characterized by the condition that $S_1S_2, S_1\vert _{\bigcap_{n\ge 0}\ker S_2^*S_1^n}$ and $S_2\vert _{\bigcap_{n\ge 0}\ker S_1^*S_2^n}$ are shifts. Moreover, $S$ contains bi-shift and modified bi-shift maximal parts.