A criterion for essential self-adjointness of a symmetric operator   defined by some infinite hermitian matrix with unbounded entries
  We shall consider a double infinite, hermitian, complex entry matrix $A=[a_{x,y}]_{x,y\in\mathbb Z}$, with $a_{x,y}^*=a_{y,x}$, $x,y\in\mathbb Z$. Assuming that the matrix is almost of a finite bandwidth, i.e. there exists an integer $n> 0$ and exponent $\gamma\in[0,1)$ such that $ a_{x,x+z}=0$ for all $z>n\langle x\rangle^{\gamma}$ and the growth of the $\ell_1$ norm of a row is slower than $|x|^{1-\gamma}$ for $|x|\gg1$, i.e. $\lim_{|x|\to+\infty}| x|^{\gamma-1}\sum_{y}|a_{xy}|=0$ we prove that the corresponding symmetric operator, defined on compactly supported sequences, is essentially self-adjoint in $\ell_2(\mathbb Z)$. In the case $\gamma=0$ (the so called $(nJ)$-matrices) we prove that there exists $c_*>0$, depending only on $n$, such that the condition $\limsup_{|x|\to+\infty}| x|^{-1}\sum_{y}|a_{xy}|\le c_*$ suffices to conclude essential self-adjointness. 
Tomasz Komorowski
math.FA