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).