We obtain a Shimorin Wold-type decomposition for a doubly commuting covariant representation of a product system of \(C^*\)-correspondences over \({\mathbb {N}}_0^k\). This result gives Shimorin-type decompositions of recent Wold-type decompositions by Jeu and Pinto (Adv Math 368:107–149, 2020) for the q-doubly commuting isometries and by Popescu (J Funct Anal 279:108798, 2020) for Doubly \(\Lambda \)-commuting row isometries. Application to the wandering subspaces of the induced representations is explored, and a version of the Beurling–Lax-type characterization is obtained to study doubly commuting invariant subspaces.

我们为\(C^*\) - 对应于\({\mathbb {N}}_0^k\)的乘积系统的双重交换协变表示获得 Shimorin Wold 型分解。该结果给出了 Jeu 和 Pinto (Adv Math 368:107–149, 2020) 对q双重交换等距线和 Popescu (J Funct Anal 279:108798, 2020) 对 Doubly的最近 Wold 型分解的 Shimorin 型分解\(\Lambda \) -交换行等距。探索了对诱导表示的游荡子空间的应用，并获得了 Beurling-Lax 型表征的一个版本，以研究双交换不变子空间。