Let $L$ be the Ornstein-Uhlenbeck operator on $L^2(\gamma)$ where $\gamma$ is the Gaussian measure on $\mathbf R^d$. Hille-Yosida or Lumer-Phillips can be used to prove that $L$ generates a strongly continuous semigroup $e^{tA}$.
Now I have the following integral:
$\int_0^\infty (t^2 L)^{N + 1} e^{\beta t^2 L} u \, \frac{\text{d}t}{t},$
where $u \in L^2$, $\beta > 0$.
This doesn't look like a normal Bochner integral. I guess I need some kind of functional calculus. I would like to know how I could have this integral make sense.