A Criterion for Essential Self-Adjointness of a Symmetric Operator Defined by Some Infinite Hermitian Matrix with Unbounded Entries

Abstract

Abstract. We shall consider a double infinite, Hermitian, complex entry matrix A = [ax,y] x,y∈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 2(Z). Roughly speaking, assuming that x denotes the row number, we require that: (1) there exist γ ∈ [0, 1) and n > 0 for which the entries that are at distance larger than n(|x| 2 +1) γ/2 from the diagonal vanish and (2) the 1 norm of the xth row grows slower that |x| γ−1 , as |x| → +∞.