A remark on generalized commutation relation and subnormality

Abstract. Tillmann [Tillmann H. G., Zur Eindeutigkeit der Losungen der quanten mechanischen vertauschungrelationen, Acta Sci. Math. (Szeged) 24 (1963), 258-270] proved that every operator \(A\) which fulfils the canonical commutation relation \(A^{*}A - AA^{*} = Id\) is an orthogonal sum of canonical creation operators. We extend this result for operators which fulfil generalized commutation relation \[A^{*}A - AA^{*}= E^2,\text{ where }EA = AE.\] In addition, some inequalities which are helpful in describing analytic vectors of operators \(A^{*}A\), where \(A\) fulfils the generalized commutation relation, are established.