Let $B$ be some Banach space and let $T:B \to B$ be linear and bounded. I write $\sigma_e$ for the essential spectrum, i.e. the set of $\lambda \in \mathbb{C}$ s.t. $T-\lambda$ is not Fredholm. The radius of the essential spectrum is for example given by the Nussbaum-Formula: $\varrho_e(T) = \limsup_{n \to \infty} \| T^n \|_{cpt} ^{1/n}$. Here, $\|T\|_{cpt} := \inf \{ \|T-K\|_{op}:K: B \to B \text{ is compact} \}$. For an element of the whole spectrum $\lambda \in \sigma(T)$, we know: $(T-\lambda)^{-1}$ is not a bounded linear map. So far, I hope this is correct.
Now my question: Suppose that for any $\lambda \in \sigma_e(T)$ we have that $T-\lambda$ is not injective. Does this tell us anything about the size of the essential spectrum? In particular, are there known variants of formulas (e.g. Lebow-Schlechter) for radii of the essential spectrum that simplify in this situation? What changes, if we even knew that $\dim (\ker T-\lambda) = \infty$? Thank you in advance!