I am currently working on a problem and I am stuck with the following issue.
For $A \in GL(n)$ and $B \in U(n)$ I am hoping that it is true that
$$ A(B-A)^{-1}B = B(B-A)^{-1}A $$
My question is whether this is indeed the case and if so what I need to look into to understand why. ( I just assume for the moment that $B - A$ has an inverse )
PS: Just to give some context I am stuck with this issue because I am playing around with the kernel that I have computed for the operator
$$ D := i \;I\frac{d}{dx} + B(x) : C^\infty_T ([0,1],\mathbb{C^m}) \to C^\infty_T ([0,1],\mathbb{C^m})$$ with boundary condition $f(1) = T f(0)$ for $T \in U(n)$ and $B$ Hermitian.
I am currently trying to show that $L_T \;f(1) = T L_T \; f(0)$ where $L_T$ is the integral operator given by the kernel that I have computed for $D$. In order for things to work out I have to show that
$$ p(1)(T-p(1))^{-1}T = T(T-p(1))^{-1}p(1)$$ where $p$ is assumed to conjugate $D$ to $-ip^{-1}Dp=D^p = \frac{d}{dx}I$