The Sobolev orthogonality and spectral analysis of the Laguerre polynomials {Ln−k} for positive integers k

For k∈N, we consider the analysis of the classical Laguerre differential expressionℓ−k[y](x)=1x−ke−x(−(x−k+1e−xy′(x))′+rx−ke−xy(x)) (x∈(0,∞)),where r⩾0 is fixed, in several nonisomorphic Hilbert and Hilbert–Sobolev spaces. of these spaces, specifically the Hilbert space L2((0,∞);x−ke−x), it is well known that the Glazman–Krein–Naimark theory produces a self-adjoint operator A−k, generated by ℓ−k[·], that is bounded below by rI, where I is the identity operator on L2((0,∞);x−ke−x). Consequently, as a result of a general theory developed by Littlejohn and Wellman, there is a continuum of left-definite Hilbert spaces {Hs,−k=(Vs,−k,(·,·)s,−k)}s>0 and left-definite self-adjoint operators {Bs,−k}s>0 associated with the pair (L2((0,∞);x−ke−x),A−k). For A−k and each of the operators Bs,−k, it is the case that the tail-end sequence {Ln−k}n=k∞ of Laguerre polynomials form a complete set of eigenfunctions in the corresponding Hilbert spaces. 5, Kwon and Littlejohn introduced a Hilbert–Sobolev space Wk[0,∞) in which the entire sequence of Laguerre polynomials is orthonormal. In this paper, we construct a self-adjoint operator in this space, generated by the second-order Laguerre differential expression ℓ−k[·], having {Ln−k}n=0∞ as a complete set of eigenfunctions. The key to this construction is in identifying a certain closed subspace of Wk[0,∞) with the kth left-definite vector space Vk,−k.