A General Left-Definite Theory for Certain Self-Adjoint Operators with Applications to Differential Equations

Abstract

DEDICATED TO W.N. EVERITT, AN INSPIRATION AND MENTOR TO BOTH AUTHORSWe show that any self-adjoint operator A (bounded or unbounded) in a Hilbert space H ¼ ðV ; ðÁ; ÁÞÞ that is bounded below generates a continuum of Hilbert spaces fH r g r>0 and a continuum of self-adjoint operators fA r g r>0 . For reasons originating in the theory of differential operators, we call each H r the rth left-definite space and each A r the rth left-definite operator associated with ðH ; AÞ. Each space H r can be seen as the closure of the domain DðA r Þ of the self-adjoint operator A r in the topology generated from the inner product ðA r x; yÞ ðx; y 2 DðA r ÞÞ. Furthermore, each A r is a unique self-adjoint restriction of A in H r . We show that the spectrum of each A r agrees with the spectrum of A and the domain of each A r is characterized in terms of another left-definite space. The Hilbert space spectral theorem plays a fundamental role in these constructions. We apply these results to two examples, including the classical Laguerre differential expression '½Á in which we explicitly find the leftdefinite spaces and left-definite operators associated with A, the self-adjoint operator generated by '½Á in L 2 ðð0; 1Þ; t a e Àt Þ having the Laguerre polynomials as eigenfunctions. # 2002 Elsevier Science (USA) Key Words: spectral theorem; self-adjoint operator; Hilbert space; Sobolev space; Dirichlet inner product; left-definite Hilbert space; left-definite self-adjoint operator; Laguerre polynomials; Stirling numbers of the second kind.Contents.

A combinatorial interpretation of the Legendre-Stirling numbers

“…824-825] and [4, Chapter V]). Indeed, in a new application of the Stirling numbers of the second kind, it is reported in [9] that the Stirling numbers of the second kind {S (j) n } are the coefficients of the integral composite powers of the second-order Laguerre differential expression:…”

“…In [9], Littlejohn and Wellman generalized left-definite theory from its traditional roots in differential equations to a more abstract setting, namely to arbitrary selfadjoint operators A that are bounded below in a Hilbert space (H, (·, ·)) by a positive constant k; that is to say,…”

“…However, as the authors show in [9] (see also [3], [6], [7], and [8]), it is possible to compute these spaces and inner products for several well-known self-adjoint operators for each positive integer r.…”

Nonclassical Jacobi Polynomials and Sobolev Orthogonality

“…In [13], Littlejohn and Wellman developed a general abstract left-definite theory for a self-adjoint operator A that is bounded below in a Hilbert space (H, (·, ·)).…”

“…More importantly, we construct a self-adjoint, positively bounded below operator T α , generated from l α,−1 [·], in W α having the entire set of Jacobi polynomials {P (α,−1) n } ∞ n=0 as a complete set of eigenfunctions. The general left-definite theory, that was recently developed by Littlejohn and Wellman [13] is, surprisingly, of paramount importance in the construction of this self-adjoint operator.…”