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.
We develop the left-definite analysis associated with the self-adjoint Jacobi operator A ( , ) k , generated from the classical secondorder Jacobi differential expressionin the Hilbert space L 2 , (−1, 1) := L 2 ((−1, 1); w , (t)), where w , (t) = (1 − t) (1 + t) , that has the Jacobi polynomials {P ( , ) m } ∞ m=0 as eigenfunctions; here, , > − 1 and k is a fixed, non-negative constant. More specifically, for each n ∈ N, we explicitly determine the unique left-definite Hilbert-Sobolev space W ( , ) n,k (−1, 1) and the corresponding unique left-definite selfadjoint operator B ( , )
Stirling numbers of the second kind Stirling numbers of the first kind Left-definite theory Combinatorics Euler criterion a b s t r a c tThe Legendre-Stirling numbers are the coefficients in the integral Lagrangian symmetric powers of the classical Legendre second-order differential expression. In many ways, these numbers mimic the classical Stirling numbers of the second kind which play a similar role in the integral powers of the classical second-order Laguerre differential expression. In a recent paper, Andrews and Littlejohn gave a combinatorial interpretation of the Legendre-Stirling numbers. In this paper, we establish several properties of the Legendre-Stirling numbers; as with the Stirling numbers of the second kind, they have interesting generating functions and recurrence relations. Moreover, there are some surprising and intriguing results relating these numbers to some classical results in algebraic number theory.
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