Hilbert modules over a class of semicrossed products (Q2706586)

This article deals with a disc algebra \({\mathcal A}(D)\) and the corresponding associated non-selfadjoint operator algebra \(\mathbb{Z}^+ \times_\alpha {\mathcal A}(D)\) \((\alpha\) is an fixed automorphism in \({\mathcal A}(D))\); this operator algebra is called the semicrossed product of \({\mathcal A}(D)\) and \(\alpha\). The main result characterizes the Shilov and orthogonally projective moduls over \(\mathbb{Z}^+ \times_\alpha {\mathcal A}(D)\) as those corresponding to pairs of isometries \(S\) and \(T\) satisfying \(TS=S \alpha(T)\). Further, some results concerned with the commutant lifting for \(\mathbb{Z}^+ \times_\alpha {\mathcal A}(D)\) are presented.