Spectral gap of the generator of a contraction semi-group

Question

Let $(P_t)_{t\ge 0}$ be a strongly continuous contraction semi-group (not necessarily self-adjoint) of operators with $P_0=I$ defined on a Hilbert Space $(H,\langle\cdot,\cdot\rangle$). Suppose $(A,D(A))$ is the generator of this semi-group. Assume that the operator $-A$ admits spectral gap, i.e, there exists $\lambda >0$ such that $\Re(\alpha)\ge\lambda$ for all $\alpha\in\sigma(-A)$, where $\sigma(-A)$ denotes the spectrum of $-A$. Is it true that $|\langle -Af,f\rangle|\ge\lambda\|f\|^2$ for all $f\in D(A)$?