Laguerre polynomial expansions

The differential operator generated by the Laguerre differential equation xy″ + (1 + α − x)y′ + λy = 0 is considered for all α both in the classical setting L2(0,∞; xαe−x) and in the Sobolev space H1(0,∞;xα + 1e−x;xαe−x). In L2 a shifted operator exists for α < 1, which corresponds to the classic case α > −1. In H1, we prove that for α ⩾ 0 the Laguerre operator, self-adjoint in L2, remains self-adjoint with a restricted domain. For α ⩽ 0 a shifted operator corresponds to the case α ⩾ 0, just as in the L2 case.