Let $H$ Hilbert, $\{e_k\}$ orthonormal basis, $A \colon D(A) \subset H \to H$ generator of a strongly continuous semigroup $e^{At}$ and $A$ such that $$Ae_k=-\lambda_k e_k$$ for some eigenvalues $\lambda_k>0$ such that $$\inf_k \lambda_k =\lambda>0$$
Then it follows by Hille-Yosida theorem that $$|e^{At}| \leq M e^{-\lambda t}$$ for some $M>0$.
Now my question is: is it true that $$|e^{At}| \leq e^{-\lambda t}$$ so that we can have $M=1$ and then say that $e^{At}$ is a semigroup of contractions?
Since $A$ is diagonalizable, the associated semigroup is given by $$e^{A t} x =\sum_{k\ge 0} e^{-\lambda_k t} \langle x,e_k\rangle e_k, \qquad \forall t\ge 0, \forall x\in H.$$ Then by Pythagoras's Theorem,
\begin{align*} |e^{At}x|^2 &=\sum_{k\ge 0} e^{-2\lambda_k t} |\langle x,e_k\rangle|^2\\ & \le e^{-2\lambda t} |x|^2, \qquad \forall x\in H. \end{align*}