Compute an operator norm 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 
$$
\int p(x)^2d\mu(x)\leq 1.
$$
 A: We consider $M_n\colon\mathbb R_n[X]\to\mathbb R[X]$ defined by $M_n(p)(x)=xp(x)$. We want to get a bound for $\lVert M_n\rVert=\sup_{\lVert p\rVert=1,p\in\mathbb R_n[X]}\lVert M_n\rVert$. We can construct $\{p_k\}$, an orthonormal sequence of polynomials such that $\operatorname{deg}p_k=k$. Let $p\in\mathbb R_n[X]$, $\lVert p\rVert=1$. We can write $p=\sum_{j=0}^na_jp_j$ where $a_j$ are real numbers, and $||p||=1$ means $\sum_{j=0}^na_j^2=1$. We have 
\begin{align*}
||M_n(p)||^2&=\int_{\mathbb R}x^2p(x)^2d\mu(x)\\
&=\sum_{i=0}^n\sum_{j=0}^na_ia_j\int_{\mathbb R}xp_i(x)xp_j(x)d\mu(x)\\
&\leq \sum_{i=0}^n\sum_{j=0}^n|a_i||a_j|\left(\int_{\mathbb R}x^2p_i(x)^2d\mu\right)^{1/2}\left(\int_{\mathbb R}x^2p_j(x)^2d\mu(x)\right)^{1/2}\\
&=\left(\sum_{j=0}^n|a_j|\left(\int_{\mathbb R}x^2p_j(x)^2d\mu\right)^{1/2}\right)^2\\
&\leq \sum_{j=0}^n|a_j|^2\sum_{j=1}^n\int_{\mathbb R}x^2p_j(x)^2d\mu\\
&=\sum_{j=0}^n\int_{\mathbb R}x^2p_j(x)^2d\mu,
\end{align*} 
which proves that $||M_n||\leq \sum_{j=0}^n\int_{\mathbb R}x^2p_j(x)^2d\mu$.
