Let T be a contraction on the Hilbert space H\documentclass[12pt]{minimal}
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\begin{document}$$\mathscr {H}$$\end{document} and S a minimal isometric dilation of T. In this paper, we show that every projection in {T}′\documentclass[12pt]{minimal}
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\begin{document}$$\{T\}'$$\end{document} can be extended to a projection in {S}′\documentclass[12pt]{minimal}
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\begin{document}$$\{S\}'$$\end{document}. Using this result, a sufficient condition for reducibility of ABnθ\documentclass[12pt]{minimal}
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\begin{document}$$A^{\theta }_{B_{n}}$$\end{document}, where Bn\documentclass[12pt]{minimal}
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\begin{document}$$B_{n}$$\end{document} is a finite Blaschke product with order n, is given. In particular, we determine when ABnθ\documentclass[12pt]{minimal}
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\begin{document}$$A^{\theta }_{B_{n}}$$\end{document} is reducible in two special cases. One case is that n=2,3\documentclass[12pt]{minimal}
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\begin{document}$$n=2,3$$\end{document} and the other case is that Bn=zn\documentclass[12pt]{minimal}
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\begin{document}$$B_{n}=z^{n}$$\end{document} (n∈N\documentclass[12pt]{minimal}
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\begin{document}$$n\in \mathbb {N}$$\end{document}) and θ\documentclass[12pt]{minimal}
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\begin{document}$$\theta $$\end{document} is a singular inner function.
