The Square of the Laplace Transform

Question

I have been looking at the Laplace transform $$\mathcal{L}f(s)=\int_0^{\infty}f(t)e^{-st}dt$$ and I'm trying to find

1. The norm of $\mathcal{L}^2$
2. The nullspace of The norm of $\mathcal{L}^2$

So far, I've proven an interesting identity involving $\mathcal{L}$ and the Gamma function $\Gamma$ and shown that $\mathcal{L}$ extends to a bounded map from $L^2(\mathbb{R}_{+})$ to itself.

Finally, I'd like to know what I can say about the inverse Laplace transform, in particular, whether or not it is bounded.

Thank you.