Suppose that $V \subset \mathcal{H} \subset V'$, where $\mathcal{H}$ is a Hilbert space and the dual $V'$ is taken with respect to the $\mathcal{H}$-pairing. Let's say $V$ is also a Hilbert space (continuously and densely included in $\mathcal{H}$, but in my case not compactly).
Suppose $A \colon \mathcal{H} \to \mathcal{H}$ is a self-adjoint operator such that $A \colon V \to V$ is also continuous. Then there is a natural extension of $A$ to $A' \colon V' \to V'$ via duality.
My question is: Suppose that $\lambda \in \mathbb{C}$ and $v \in V'$ is such that $A'v = \lambda v$. Does it follow that $\lambda$ belongs to the spectrum of $A \colon \mathcal{H} \to \mathcal{H}$?
In the setting I am interested in, $V$ and $\mathcal{H}$ are weighted $L^2$-spaces. I'm interested in getting a positive result -- I don't see how to arrive at one at all -- so adding hypotheses is OK. However, the Gelfand setting where $V$ is a nuclear space seems very difficult to set up in practice. But even in the setting with a nuclear space I do not see how the argument would go.