A Wold-type decomposition for commuting isometric pairs

Abstract

We obtain a Wold-type decomposition theorem for an arbitrary pair of commuting isometries V V on a Hilbert space. More precisely, V 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 ) S=(S_1,S_2) that can be characterized by the condition that S 1 S 2 ,   S 1 | ⋂ n ≥ 0 ker ⁡ S 2 ∗ S 1 n S_1S_2,\ S_1|_{\bigcap _{n\ge 0}\ker S_2^*S_1^n} and S 2 | ⋂ n ≥ 0 ker ⁡ S 1 ∗ S 2 n S_2|_{\bigcap _{n\ge 0}\ker S_1^*S_2^n} are shifts. Moreover, S S contains bi-shift and modified bi-shift maximal parts.