On self-adjoint operators generated by the fourth-order Laguerre type differential expression

Abstract

In this paper we address the problem of finding all self-adjoint operators generated by a Lagrangian symmetrizable differential expression in a Hilbert space with a mixed Sobolev inner product strictly discrete on the derivatives. We start giving survey on the initial results in the literature: the Glazman-Krein-Naimark (GKN) theory, the later developed GKN-EM theory as an extension of the first, and the works of Littlejohn and Wellman on the subject.We apply those to the fourth-order Laguerre type differential expression known for having a sequence of orthogonal polynomials as eigenfunctions and a symmetrizable differential expression. As a result, we give a complete description of the operator space defined from the GKN-EM theory, and also a characterization of all possible self-adjoint operators in the Sobolev space where the operator is defined. We validate these results with known examples previously studied in the literature.