Extensions of Laguerre operators in indefinite inner product spaces

The Laguerre–Sonin polynomials $L_n^{(\alpha)}$ are orthogonal in linear spaces with indefinite inner product if $\alpha<-1$. We construct the completion $\Pi(\alpha)$ of this space and describe self-adjoint extensions of the Laguerre operator $l(y)=xy''+(1+\alpha-x)y'$, $\alpha<-1$, in the space $\Pi(\alpha)$. In particular, we write out the self-adjoint extension of the Laguerre operator whose eigenfunctions coincide with the Laguerre–Sonin polynomials and form an orthogonal basis in $\Pi(\alpha)$.