A General Left-Definite Theory for Certain Self-Adjoint Operators with Applications to Differential Equations
A General Left-Definite Theory for Certain Self-Adjoint Operators with Applications to Differential Equations
复制标题
某些自伴算子的一般左定理论及其在微分方程中的应用
DOI：
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发表时间：
2002
期刊：
影响因子：
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通讯作者：
R. Wellman
中科院分区：
文献类型：
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作者：
L. Littlejohn;R. Wellman
我们证明了Hilbert空间H=（V,（·，·））下有界的任意自伴随算子A（有界或无界）生成Hilbert空间{Hr}r>和自伴随算子{Ar}r>0的连续统。由于源自微分算子理论的原因，我们称每个Hr为第n个左定空间，称每个Ar为与（H，A）相关的第n个左定算子。每个空间Hr都可以看作是由内积（Arx,y） （x,y∈D(Ar)）生成的拓扑中自伴随算子Ar的定义域D（Ar）的闭包。此外，每个Ar都是a在Hr中的唯一自伴随约束。我们证明了每个Ar的谱与A的谱一致，并且每个Ar的定义域用另一个左定空间表示。希尔伯特空间谱定理在这些构造中起着基本的作用。我们将这些结果应用到两个例子中，包括经典的Laguerre微分表达式l[·]，其中我们显式地找到了与A相关的左定空间和左定算子，即l[·]在L2(（0,∞）中产生的自伴随算子；tαe−t)有拉盖尔多项式作为特征函数。
We show that any self-adjoint operator A (bounded or unbounded) in a Hilbert space H=(V,(·,·)) that is bounded below generates a continuum of Hilbert spaces {Hr}r>0 and a continuum of self-adjoint operators {Ar}r>0. For reasons originating in the theory of differential operators, we call each Hr the rth left-definite space and each Ar the rth left-definite operator associated with (H,A). Each space Hr can be seen as the closure of the domain D(Ar) of the self-adjoint operator Ar in the topology generated from the inner product (Arx,y) (x,y∈D(Ar)). Furthermore, each Ar is a unique self-adjoint restriction of A in Hr. We show that the spectrum of each Ar agrees with the spectrum of A and the domain of each Ar 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 l[·] in which we explicitly find the left-definite spaces and left-definite operators associated with A, the self-adjoint operator generated by l[·] in L2((0,∞);tαe−t) having the Laguerre polynomials as eigenfunctions.