Let $F(z)$ be an $n\times m$ ($n\ge m$) rational matrix-valued function such that the column rank of $F(z)$ is full a.e. in $z\in\mathbb{C}$. Let $X$ and $Y$ be two complex constant $m\times n$ matrices. Further suppose that:
- $F(z)$ is element-wisely bounded and analytic on the complement of the unit disk in the complex plane.
- $\lim_{z\to \infty}XF(z)$ is lower-triangular with (strictly) positive entries on the diagonal and $\lim_{z\to \infty}YF(z)$ is lower-triangular with real entries on the diagonal.
Question: Does the following condition $$ \frac{1}{2\pi}\int_{-\pi}^{\pi}(YF(e^{i\theta}))(XF(e^{i\theta}))^{-1} \mathrm{d} \theta = 0 $$ imply that $Y=0$?
A simple remark. If $n=m$ it is easy to see that the above condition implies $YX^{-1} = 0$, which yields the thesis.