A map $U(\theta)$ is said to be analytic at $\theta_0$ if there exist bounded operators $U_n$ such that we have
$$U(\theta)=\sum(\theta-\theta_0)^nU_n$$
Let $p(x)$ be a polynomial in $R^n$. Consider the set $A$ of function of the type $$\psi(x)=p(x)e^{-\alpha x^2}$$
where $\alpha >0$
This set is dense in $L^2(R^n)$
Let $U(\theta)$ be such that $$U(\theta)\psi(x)=e^{\frac{n \theta}{2}}p(xe^{\theta})e^{-\alpha e^{2 \theta}x^2}$$ where $\theta$ is a complex number.
How to proof that $U(\theta)$ is analytic and that real of $U(\theta)\psi(x)$ is dense in $A$?