Confusion about defect numbers and resolvent

Question

Suppose $H$ is a complex Hilbert space and $A:D(A)\to H$ is a densely defined, closed operator. Then consider the defect number $d_{\lambda}(A):=\text{dim}(\text{ran}(T-\lambda)^{\perp})$ of $A$ at $\lambda\in \Pi(A):=\{\lambda\in\mathbb{C}: A-\lambda\text{ has a bounded inverse on its range}\}$. I have the result that in this case the resolvent set $\rho(A)=\{\lambda\in\Pi(A): d_\lambda(A)=0\}$. As far as I can see, this is the same as saying that $\text{ran}(T-\lambda)$ is dense. The spectrum is the complement of the resolvent set, yet the continuous part of the spectrum consists of those $\lambda\in\mathbb{C}$ for which $T-\lambda$ is injective, not surjective but has dense range. This seems to be a contradiction. What am I missing?