Stability of semigroups commuting with a compact operator

Abstract:

It is proved that if $T(t), S(t)$ are bounded $C_{0}$-semigroups on Banach spaces $X$ and $Y$, resp., and $C:Y\to X$, $K:Y\to Y$ are bounded operators with dense ranges such that $C$ intertwines $T(t)$ with $S(t)$ and $K$ commutes with $S(t)$, then $T(t)$ is strongly stable provided $A$—the generator of $T(t)$—does not have eigenvalue on $i\mathbf {R}$. An analogous result holds for power-bounded operators.