Hilbert modules over a class of semicrossed products

Given the disk algebra $\mathcal{A}(\mathbb{D})$ and an automorphism $\alpha$, there is associated a non-self-adjoint operator algebra $\mathbb{Z}^+\times_{\alpha}\mathcal{A}(\mathbb D)$ called the semicrossed product of $\mathcal{A}(\mathbb{D})$ with $\alpha$. Buske and Peters showed that there is a one-to-one correspondence between the contractive Hilbert modules $\mathcal{H}$over $\mathbb{Z}^+\times_{\alpha}\mathcal{A}(\mathbb D)$and pairs of contractions $S$ and $T$ on $\mathcal{H}$satisfying $TS=S\alpha(T)$. In this paper, we show that the orthogonally projective and Shilov Hilbert modules $\mathcal{H}$ over $\mathbb{Z}^+\times_{\alpha}\mathcal{A}(\mathbb D)$correspond to pairs of isometries on $\mathcal{H}$ satisfying $TS=S\alpha(T)$. The problem of commutant lifting for $\mathbb{Z}^+\times_{\alpha}\mathcal{A}(\mathbb D)$ is left open, but some related results are presented.