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

TL;DR

This paper establishes criteria for the essential self-adjointness of symmetric operators defined by certain infinite hermitian matrices with unbounded entries, focusing on matrices with almost finite bandwidth and controlled row growth.

Contribution

It provides new sufficient conditions for essential self-adjointness of operators associated with infinite matrices under specific decay and bandwidth assumptions.

Findings

Proves essential self-adjointness under almost finite bandwidth conditions.

Identifies a specific bound for $(nJ)$-matrices ensuring self-adjointness.

Extends previous results to matrices with unbounded entries and controlled growth.

Abstract

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.