Subinner-free outer factorizations on an annulus

Recent work of Aleman, Hartz, McCarthy and Richter generalizes the classical inner-outer factorization of Hardy space functions to the complete Pick space setting, establishing an essentially unique “subinner-free outer” factorization. In this note, we investigate certain special examples of such factorizations in the setting of the function space induced on the annulus $A_r=\{r<\mathrm{\mid }z\mathrm{\mid }<1\}$ by the complete Pick kernel $$k_r(\lambda ,\mu ):=\frac{1-r^2}{(1-\lambda \bar{\mu })(1-r^2\mathrm{/}\lambda \bar{\mu })}\mathrm{.}$$