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 $A_r=\{r<|z|<1\}$ by the complete Pick kernel $$k_{r}(\lambda,\mu):=\frac{1-r^2}{(1-\lambda\bar{\mu})(1-r^2/\lambda\bar{\mu})}.$$