The differential operator generated by the Laguerre differential equationxy + (1 + α - x)y + λy = 0L 2 (0, ∞; x α e - x ) and in the Sobolev spaceH 1 (0, ∞; x α + 1 e - x ;x α e - x ). InL 2 a shifted operator exists for α < 1, which corresponds to the classic case α > -1. In H 1 , we prove that for α 0 the Laguerre operator, self-adjoint in L 2 , remains self-adjoint with a restricted domain. For α 0 a shifted operator corresponds to the case α 0, just as in the L 2 case.