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

For k is an element of N, we consider the analysis of the classical Laguerre differential expression l (-k)[y] (x) = 1/(x(-k) e(-x)) over bar (-(x (-k+1) e(-x) y' (x))' + rx(-k) e(-x) y(x)) (x is an element of(0, infinity)), where r greater than or equal to 0 is fixed, in several nonisomorphic Hilbert and Hilbert-Sobolev spaces. In one of these spaces, specifically the Hilbert space L-2 ((0, infinity); x(-k) e(-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 L-2 ((0, infinity); x(-k) e(-x)). Consequently, as a result of a general theory developed by Littlejohn and Wellman, there is a continuum of left-definite Hilbert spaces {H-s,H- (-k) = (V-s,V- -k,V- ((.),(.))(s, -k) )}(s>0) and left-definite self-adjoint operators {B-s,B-k}(s>0) associated with the pair (L-2((0, infinity); x(-k) e(-x)),A(-k),). For A(-k) and each of the operators Bs-k, it is the case that the tail-end sequence {L-n(-k)}(n=k)(infinity) of Laguerre polynomials form a complete set of eigenfunctions in the corresponding Hilbert spaces. In 1995, Kwon and Littlejohn introduced a Hilbert-Sobolev space W-k[0, infinity) 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 {L-n(-k)}(n=0)(infinity) as a complete set of eigen functions. The key to this construction is in identifying a certain closed subspace of W-k [0, infinity) with the kth left-definite vector space V-k,V--k. (C) 2004 Published by Elsevier B.V .