Let $H$ be a Hilbert space and $N: D(N)\subset H \to H$ be some self-adjoint operator with domain $D(N)$ dense in $H$ and spectrum $$\sigma(N) \subset (-\infty, \alpha]$$ for some fixed $\alpha <0.$ We assume further that $$e^{tN}(D(N))\subset D(N), \quad \forall t\in (0, 1].$$ Let now $g: H \to H$ such that $g(D(N)) \subset D(N)$ and there exist $C,s>0$ $$\|g(u)\| \leq C\|u\|^s\, \forall u \in H. $$ Define $$\psi(u):= \int_0^\infty e^{tN}g(tu)\, dt.$$
- Can we prove that $\psi (D(N))\subset D(N)$ and that $$N\psi(u)= \int_0^\infty N e^{tN}g(tu)\, dt ?$$
- If the answer on the above question is yes, can this result be generalised by taking $N$ to be normal with spectrum $$\sigma(N) \subset \{\lambda \in \mathbb{C}: {\rm Re}(\lambda) \leq \alpha\}?$$ The assumption on $e^{tN}$ on the interval $(0,1]$ implies that $$e^{tN}(D(N))\subset D(N), \quad \forall t\in (0, \infty).$$ Then I don't know if it makes sense to write $$\psi(u) = \sum_{n =O}^{\infty}\int_{n}^{n+1} e^{tN}g(tu)\, dt$$
Thank you for any hint.