Let us consider the space $L_2(\mathbb{R} \times [0,1]; \mathbb{R}^n)$, i.e functions taking values in $\mathbb{R}^n$ and in $L_2$ . Suppose $T$ is a bounded linear operator defined as follows:
$Tf(x,y)=M(x,y)f(x+y,y)$, where $M(x,y)$ is an $n \times n$ matrix.
Suppose we also know that $M(x,y) \to M_\infty$ as $|x| \to \infty$ for every $y \in [0,1]$ where $M_\infty$ is a constant matrix. Suppose the spectral radius of $M_{\infty} > 1$.
$M(x,y)$ also has the properties $M(x,0)=M_\infty$ and $M(x,y)=M(ax,ay)$ for $a>0$ and such that $ay \in [0,1]$
Can we prove something like the spectral radius of $T > 1$
Thank you.