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 $${A=[a_{x,y}]_{x,y\in\mathbb{Z}}}$$ . 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 $${|x|\to+\infty}$$ . [ABSTRACT FROM AUTHOR]