Extensions of Laguerre operators in indefinite inner product spaces

The Laguerre-Sonin polynomialsLn(α) are orthogonal in linear spaces with indefinite inner product ifα<−1. We construct the completion Π(α) of this space and describe self-adjoint extensions of the Laguerre operatorl(y)=xy″+(1+α−x)y′,α<−1, in the space Π(α). 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 Π(α).