A general left-definite theory for certain self-adjoint operators with applications to differential equations

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:181

Article

Littlejohn, LL ; Wellman, R

关键词: 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;   

DOI  :  10.1006/jdeq.2001.4078

来源: Elsevier

【 摘 要 】

We shock that any self-adjoint operator A (bounded or unbounded) in a Hilbert space H W, (,)) that is bounded below generates a continuum of Hilbert spaces {H-r}(r>0) and a continuum of self-adjoint operators {A(r)}(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 (H,A). Each space H-r can be seen as the closure of the domain L(A(r)) of the self-adjoint operator A(r) in the topology generated from the inner product (A(r)x, y) (x, y epsilon L(A(r))). Furthermore, each A(r) is a unique self-adjoint restriction of A in H,. 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 ([.] in which we explicitly find the left-definite spaces and left-definite operators associated with A, the self-adjoint operator generated by l[(.)] in L-2((0, infinity); t(alpha)e(-t)) having the Laguerre polynomials as eigenfunctions. (C) 2002 Elsevier Science (USA).

【 授权许可】

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