Positive Operator: Norm Estimate In class we encountered the statement:
$$H\geq C1\quad(C>0)\implies\|\mathrm{e}^{-\beta H}\|<1\quad(\beta>0)$$
How does one prove this?
Moreover, what about the weakened version:
$$H\geq C1\quad(C\geq0)\implies\|\mathrm{e}^{-\beta H}\|<1\quad(\beta\geq0)$$

You can get the result you want using only $C_{0}$-semigroup theory. Because $H \ge C > 0$, then $K=-H+CI \le 0$ is the generator of a contractive $C_{0}$-semigroup $e^{tK}=e^{tC}e^{-tH}$. So $\|e^{tC}e^{-tH}\|\le 1$, which implies $\|e^{-tH}\| \le e^{-tC}$ for all $t \ge 0$.
