I am interested in the following problem: I have an infinite symmetric tridiagonal matrix
$$
A=
\begin{bmatrix}
a_1 & b_1 & & & \\
b_1 & a_2 & b_2 & & \\
& b_2& a_3 & b_3 & \\
& & \ddots & \ddots & \ddots & \\
\end{bmatrix}
$$
where $a_j, b_j>0$, and I need to determine whether $A$ is *positive definite*, meaning that the corresponding quadratic form is bounded below:
$$
Q_A(\beta_1, \beta_2\ldots \beta_n\ldots)\stackrel{\mathrm{def}}{=}\sum_{j=1}^\infty a_j \beta_j^2 + 2b_j\beta_{j}\beta_{j+1} \ge c\sum_{j=1}^\infty \beta_j^2.$$
Here $c>0$. (If $c=0$, we say that $A$ is positive *semidefinite*).

QuestionAre there infinite-dimensional versions of the familiar criterions of linear algebra, such as the Sylvester's criterion or the diagonal dominance sufficient condition?

Any result or reference is gladly welcome.