Let $\mathcal{H}$ be a complex Hilbert space with inner product $\langle\cdot|\cdot \rangle$.
An operator $T$, with domain $D(T)$, is said accretive if $\Re\langle T x | x \rangle\geq 0$, for all $x\in D(T)$.
Now assume that $T$ is closed. A general property of $T$ concerns its resolvent set $\rho(T)$: if $\lambda \in \rho(T)$ with $\Re\lambda <0$, then $\{\lambda \in \mathbb{C}:\Re \lambda <0\}\subseteq \rho(T)$. Moreover, if $\{\lambda \in \mathbb{C}:\Re \lambda <0\}\cap \rho(T)=\varnothing$ then $\{\lambda \in \mathbb{C}:\Re \lambda <0\}$ is contained in the residual spectrum of $T$.
Is there a closed accretive operator $T$ with $D(T)$ dense in $\mathcal{H}$ such that $\{\lambda \in \mathbb{C}:\Re \lambda <0\}\cap \rho(T)=\varnothing$, but there exist $\lambda\in \rho(T)$ and $x\in D(T)$ with $\langle T x | x \rangle =\lambda \langle x|x \rangle$?