Consider an unbounded self-adjoint and strictly positive operator $A\:\mathcal{D}(A)\to\mathcal{H}$. With strictly positive, I mean $\sigma(A)\subset [a,\infty)$ for some $a>0$. Now, with the spectral theorem, I get a bounded operator $e^{-\varepsilon A}:\mathcal{H}\to\mathcal{H}$ for $\varepsilon>0$.
Does $e^{-\varepsilon A}$ preserves the support $\mathcal{D}(A^{1/2})$?
The operator $A^{1/2}$ is also defined with spectral theorem. Let $\omega\in\mathcal{D}(A^{1/2})$. I have to show that
$$\int_{\sigma(A)}\lambda\,\mathrm{d}\langle P(\lambda)e^{-\varepsilon E}\omega,e^{-\varepsilon E}\omega\rangle<\infty$$
where $P$ is the spectral measure of $A$. I wanted to use a similar approach as in the answer in this question here, but it is slightly different in my case since the real exponential yields
$$\langle P(\lambda)e^{-\varepsilon E}\omega,e^{-\varepsilon E}\omega\rangle=\langle P(\lambda)e^{-2\varepsilon E}\omega,\omega\rangle$$