Left-definite theory with applications to orthogonal polynomials

In the past several years, there has been considerable progress made on a general leftde…nite theory associated with a self-adjoint operator A that is bounded below in a Hilbert space H; the term ‘left-de…nite’has its origins in di¤erential equations but Littlejohn and Wellman [25] generalized the main ideas to a general abstract setting. In particular, it is known that such an operator A generates a continuum fHrgr>0 of Hilbert spaces and a continuum of fArgr>0 of selfadjoint operators. In this paper, we review the main theoretical results in [25]; moreover, we apply these results to several speci…c examples, including the classical orthogonal polynomials of Laguerre, Hermite, and Jacobi. 1. Introduction In this paper, we bring together several recent results, and examples, concerning the theory of self-adjoint operators that are bounded below in a Hilbert space. The mathematical literature has numerous examples of self-adjoint di¤erential operators that are bounded below in a Hilbert space that generate a ‘left-de…nite’study; a speci…c explanation of this terminology is given below in Section 2. The origins of left-de…nite theory (the ideas can be traced to fundamental work of Hermann Weyl [41] on his landmark study of second-order di¤erential equations) and the term left-de…nite (actually, the German Links-de…nit) …rst appeared in the literature in 1965 in a paper by Schäfke and Schneider [35]. Over the past forty years, there has been a resurgence in this study by several authors. In fact, there are other interpretations of ‘left-de…nite’in the literature; we refer to the paper [18] and the references cited therein for another viewpoint on ‘left-de…nite’problems. A general left-de…nite theory for arbitrary self-adjoint operators A that are bounded below in a Hilbert space was developed by Littlejohn andWellman in [25]; we refer the reader to this contribution for a detailed list of references. Since the publication of [25] in 2002, there have been several papers written on the applications of this theory to speci…c operators studied in mathematical physics and functional analysis. In particular, this theory can be applied to the second-order di¤erential equations of Laguerre, Hermite, and Jacobi which have classical orthogonal polynomial solutions. Furthermore, as a consequence of this general theory, there are new applications to combinatorics as well as new information on various powers of A: Date : Februrary 16, 2008 (C:nSWDocsnPapersnLeftdef_surveyFV.tex). 1991 Mathematics Subject Classi…cation. Primary 34B30, 47B25, 47B65; Secondary 33C65, 34B20. Key words and phrases. self-adjoint operator, Hilbert space, Sobolev space, Dirichlet inner product, left-de…nite Hilbert space, left-de…nite self-adjoint operator, Laguerre polynomials, Stirling numbers of the second kind, LegendreStirling numbers, Jacobi-Stirling numbers. This paper is based on a plenary lecture given by L. L. Littlejohn in honor of Professor J. S. Dehesa at the conference entitled "Special functions, Information theory, Mathematical physics" in Granada, Spain from September 17-19, 2007. 1 2 ANDREA BRUDER, LANCE L. LITTLEJOHN, DAVUT TUNCER, AND R. WELLMAN The contents of this paper are as follows. In Section 2, we motivate this left-de…nite theory from the original viewpoint of di¤erential equations. In Section 3, we review the main results appearing in [25]. Lastly, in Section 4, we consider several examples to illustrate this theory. The main aim of this paper is to ‘draw’further mathematicians into the subject of left-de…nite theory. Indeed, there are numerous self-adjoint ordinary, partial and di¤erence equations for which this theory may be applied. As the reader will see in this paper, considerable more information can be derived about the original self-adjoint operator by considering the associated left-de…nite theory. 2. Left-Definite Theory from the Viewpoint of Differential Equations As mentioned in the Introduction, the left-de…nite theory has its origins in di¤erential equations; we now discuss this connection in more detail. For the sake of clarity and simplicity, we consider di¤erential expressions that have smooth coe¢ cients but note that more general expressions can be considered. We will also limit our discussion to motivating …rst left-de…nite spaces and …rst leftde…nite operators; indeed, until the paper [25] appeared, there was no mention of other left-de…nite spaces or operators in the literature. Let `[ ] be the di¤erential expression (2.1) `[y](x) = 1 w(x) N X j=0 ( 1) aj(x)y (x) (j) (x 2 I);