A criterion for essential self-adjointness of a symmetric operator defined by some infinite hermitian matrix with unbounded entries

Abstract

We shall consider a double infinite, hermitian, complex entry matrix $A=[a_{x,y}{]}_{x,y\in \mathbb{Z}}$, with $a_{x,y}^{\ast }=a_{y,x}$, $x,y\in \mathbb{Z}$. 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 $z>n⟨x{⟩}^{\gamma }$ and the growth of the ${\ell }_1$ norm of a row is slower than $|x{|}^{1-\gamma }$ for $|x|\gg 1$, 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 $c_{\ast }>0$, depending only on $n$, such that the condition ${\mathrm{limsup}}_{|x|\to +\infty }|x{|}^{-1}{\sum }_y|a_{xy}|\le c_{\ast }$ suffices to conclude essential self-adjointness.