Let $\Phi \hookrightarrow \mathcal{H} \hookrightarrow \Phi^*$ be a Gelfand triple, $\Phi$ a nuclear space that lies dense in $\mathcal{H}$ and $A : \Phi \to \Phi$ be a linear and continuous operator that has a unique self-adjoint extension $\bar{A}$ to a set $\mathcal{D} \left( \Phi \right) \supseteq \Phi$. Then, according to Blanchard, Brüning, $\bar{A}$ has a complete set of generalized eigenvectors in $\Phi^*$.
I am looking for a good way to start with a continuous symmetric bilinear form $b$ on $\Phi \times \Phi$ instead. It seems that one has to require $b$ to be extendable to a (separately?) continuous bilinear form on $\Phi \times \mathcal{H}$ in order to get things to work. This seems rather unelegant unsymmetric though. Are there better ways? I am rather inexperienced in working with nuclear spaces.