Consider the operator $M$ acting on the space $\mathbb{R}[X]$ of real polynomials by $Mp(x)=xp(x)$. We equip $\mathbb R[X]$ with the $L^2$ norm $ \|p\|^2=\int p(x)^2d\mu(x), $ where $\mu$ is a Radon measure having all its moments, namely $\mathbb R[X]\subset L^2(\mu)$. The question is : Can we find a nice formula for the operator norm $\|M\|_{op}$ ?
EDIT : The answer may be $+\infty$ for some $\mu$, see the comment below from Davide Giraudo.
In order to have a chance to obtain a formula in this general setting, I change the question by asking if one can find a ($n,\mu$-dependent) expression for $ \sup \int x^2p(x)^2d\mu(x) $ where the supremun is taken over all polynomials of degree less that $n$ satisfying $ \int p(x)^2d\mu(x)\leq 1. $