Let $A$ be a matrix satisfying $A^*+A\leq0$, it can be shown that $\|e^{tA}\|_2\leq1$ for all $t\geq 0$, where $\|\cdot\|_2$ is the spectral norm defined as largest singular value of the matrix.
I am curious if there is a theorem describing the spectral norm of the truncated exponential. Define $p_n(t) = I + tA + \frac{t^2}{2!}A^2 + ...+ \frac{t^n}{n!}A^n $. Is there a general theorem stating that, for what value $n$ and under what constraints for $t$ (for example, $t\|A\|_2\leq c_n$ for some constant $c_n$), as long as $A^*+A\leq0$, we still have $\|p_n(t)\|_2\leq1$?