I will formulate this question in the language of Jacobi operators and spectral measures although it could be entirely rewritten in terms of orthogonal polynomials and measures of orthogonality.

** Objects of interest:**
Let $J$ be a semi-infinite Jacobi matrix of the form
$$J=\begin{pmatrix}
b_{1} & a_{1}\\
a_{1} & b_{2} & a_{2}\\
& a_{2} & b_{3} & a_{3}\\
& & \ddots & \ddots & \ddots
\end{pmatrix},$$
where $b_{n}\in\mathbb{R}$ and $a_{n}>0$. Let $\mathcal{M}$ denotes the set of Jacobi matrices $J$ such that $b_{n}\to 0$ and $a_{n}\to 1$ as $n\to\infty$ (the so-called Szegö class in OG polynomials terminology).

Let $J_{0}$ be the Jacobi matrix with $b_{n}=0$ and $a_{n}=1$ for all $n\in\mathbb{N}$. It is well-known that the spectrum of $J_{0}$ is $$\sigma(J_{0})=[-2,2].$$ A Jacobi matrix $J\in\mathcal{M}$ iff $J-J_{0}$ is compact.

For any self-adjoint operator $J$ (assume $J$ is bounded), there exists unique measure $\mu$ (the spectral measure) determined by the equality $$\int_{\mathbb{R}}\frac{d\mu(x)}{x-z}=\langle e_{1},(J-z)e_{1}\rangle, \quad \forall z\in\mathbb{C}\setminus\mathbb{R},$$ where $e_{1}$ is the first vector of the standard basis of $\ell^{2}(\mathbb{N})$. It holds that $\mbox{supp}\mu=\sigma(J)$.

**Question:** Assume $\mu$ to be a probability measure with $\mbox{supp}\mu=[-2,2]$. Then there exists a unique Jacobi operator $J$ whose spectral measure coincides with $\mu$. I am interested in what additional conditions (to $\mbox{supp}\mu=[-2,2]$) is to be imposed on the measure $\mu$ to guarantee that $J\in\mathcal{M}$. The condition (if any) should be weaker than the Szegö condition which gives us more, see below.

**What is known:** There is an answer to this kind of question known for a long time but it implies even more than just $J\in\mathcal{M}$. Namely, if $\mbox{supp}\mu=[-2,2]$ and the density $\rho$ of the the Lebesgue
absolutely continuous component of $\mu$ satisfies the so-caled Szegö condition:
$$\int_{-2}^{2}\frac{\log\rho(x)}{\sqrt{4-x^{2}}}dx>-\infty,$$
then
$$\sum_{n=1}^{\infty}b_{n}^{2}<\infty \quad \mbox{ and } \quad
\sum_{n=1}^{\infty}(a_{n}-1)^{2}<\infty.$$
In particular, $J\in\mathcal{M}$. Proof and references for these claims (and much more) can be found here.