# Uniform boundedness of resolvents on the imaginary axis

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.

• While the answer is, in general, negative (see Michael Renardy's answer), a sufficient condition for a positive answer is that $\mathbb{H}$ is an $L^2$-space and the semigroup is positive; see for instance Corollary C-III-1.3 on page 294 in "Arendt et. al.: One-parameter Semigroups of Positive Operators (1986)". – Jochen Glueck May 14 at 7:50
• By the way, a rather comprehensive overview about the relation between spectral bounds, pseudo-spectral bounds and growth bounds of $C_0$-semigroups can be found in Secctions 5.1 and 5.2 of "Arendt, Batty, Hieber, Neubrander: Vector-valued Laplace Transforms and Cauchy Problems (2011)". Using the notation from this book, your question is whether $s(A) = s_0(A)$, and an explicit counterexample to this is given in Example 5.1.10 (according to the notes at the end of Chapter 5, this example is a modification, due to Wrobel, of Zabczyk's counterexample mentioned in Michael Renardy's answer. – Jochen Glueck May 14 at 8:02