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 \({A=[a_{x,y}]_{x,y\in\mathbb{Z}}}\). In the present note we give a criterion, expressed in terms of the entries of the matrix, for the corresponding symmetric operator defined on compactly supported sequences, to be essentially self-adjoint in the space \({\ell_2(\mathbb{Z})}\). Roughly speaking, assuming that x denotes the row number, we require that: (1) there exist \({\gamma\in[0,1)}\) and n > 0 for which the entries that are at distance larger than \({n(|x|^2+1)^{\gamma/2}}\) from the diagonal vanish and (2) the \({\ell^1}\) norm of the xth row grows slower that \({|x|^{\gamma-1}}\), as \({|x|\to+\infty}\).