I am having trouble with the following problem:
a) Let $H_n (x), n \in \mathbb{N}$ be Hermite polynomials associated to the measure $$ d \mu = \frac{1}{\sqrt{2\pi}} e^{-\frac{1}{2}x^2}d \lambda $$ Prove that the following operator is contractive in $\mathbb{L}^2(\mathbb{R}, \mu)$ for $\zeta \le 1$: $$T_{n}f(x) = H_k(x) + \zeta H_n(x)\int H_n(y)f(y) \mu(dy) $$
The solution is as follows: $$\|T_nf(x)-T_n g(x)\|^2 = \int|\zeta|^2 H_n(x)^2 |\int H_n(y)(f(y)-g(y)) \mu(dy)|^2 \mu(dx)$$ and hence $$ \|T_nf - T_ng\| = |\zeta| \|f-g\|$$ My question:
- How do they get from the first line to the second line? I believe it is something to do with some property of the measure but I don't see how its done. I've been staring at it for a while. If anyone could spell out the argument I would really appreciate it.
Thank you.