If $T-\lambda I$ is a Fredholm operator and $\lambda \in \sigma(T)$, then $\lambda \in \sigma_p(T)$ or $\overline{\lambda} \in \sigma_p(T^*)$?

Question

Let $H$ be a Hilbert space and let $T: D(T) \subseteq H \to H$ be a densely defined operator. Is the following statement true?

If $T-\lambda I$ is a Fredholm operator (an operator is Fredholm if its range is closed and both its kernel and its cokernel are finite-dimensional) and $\lambda \in \sigma(T)$, then $\lambda$ is an eigenvalue of $T$ or $\overline{\lambda}$ is an eigenvalue of $T^*$.

Thanks in advance for any help.