On unitary equivalence and reducing subspaces of analytic Toeplitz operator on vector-valued Hardy space

In this paper, we proved that $T_{z^n}$ acting on the ${\mathbb{C}}^m$-valued Hardy space $H_{{\mathbb{C}}^m}^2(\mathbb{D})$, is unitarily equivalent to ${⨁}_1^{mn}{T}_z$, where $T_z$ is acting on the scalar-valued Hardy space $H_{\mathbb{C}}^2(\mathbb{D})$. And using the matrix manipulations combined with operator theory methods, we completely describe the reducing subspaces of $T_{z^n}$ on $H_{{\mathbb{C}}^m}^2(\mathbb{D})$.