Let $A \colon \mathcal{D}(A) \subset \mathbb{H} \to \mathbb{H}$ be a closed linear operator in a Hilbert space $\mathbb{H}$, which generates a $C_{0}$-semigroup. Suppose that in a $\varepsilon$-neighbourhood of the imaginary axis the operator $A$ does not have spectrum. Is it true that the resolvents $(A-i\omega I)^{-1}$ are bounded in $\mathcal{L}(\mathbb{H})$ uniformly for $\omega \in \mathbb{R}$? Can the same be said for the boundedness in the norm of $\mathcal{L}(\mathbb{H};\mathcal{D}(A))$, where $\mathcal{D}(A)$ is endowed with the graph norm. If not, what are additional conditions to ensure the boundedness.

For example, if $A$ generates an exponentially stable $C_{0}$-semigroup then the boundedness of resolvents follows from the resolvent estimate in the Hille-Yosida theorem.