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 \({|x|\to+\infty}\).
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The author acknowledges the support of the Polish National Science Center grant DEC-2012/07/B/SR1/03320.
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Komorowski, T. A Criterion for Essential Self-Adjointness of a Symmetric Operator Defined by Some Infinite Hermitian Matrix with Unbounded Entries. Integr. Equ. Oper. Theory 83, 231–242 (2015). https://doi.org/10.1007/s00020-015-2237-2
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DOI: https://doi.org/10.1007/s00020-015-2237-2