On the Spectra of Left-Definite Operators
On the Spectra of Left-Definite Operators
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关于左定算子的谱
DOI：
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发表时间：
2013
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
如果A是希尔伯特空间H中下界的自伴算子，Littlejohn和Wellman（J Diff Equ 181（2）：280-339，2002）证明了，对于每个r > 0，存在唯一的希尔伯特空间Hr和Hr中唯一的自伴算子Ar，满足依赖于H和A的某些条件。空间Hr和算子Ar分别称为与（H，A）相关联的第r个左定空间和第r个左定算子。本文证明了算子A，Ar和As（r，s > 0）是等距同构等价的，空间H，Hr和Hs（r，s > 0）是等距同构的.然后利用这些结果重新构造了左定空间和左定算子。此外，我们将看到，我们的新结果意味着A和Ar的光谱是相等的，这给了我们Littlejohn和Wellman首次建立的这种现象的另一个证明（J Diff Equ 181（2）：280-339，2002）。
If A is a self-adjoint operator that is bounded below in a Hilbert space H, Littlejohn and Wellman (J Diff Equ 181(2):280–339, 2002) showed that, for each r > 0, there exists a unique Hilbert space Hr and a unique self-adjoint operator Ar in Hr satisfying certain conditions dependent on H and A. The space Hr and the operator Ar are called, respectively, the rth left-definite space and rth left-definite operator associated with (H, A). In this paper, we show that the operators A, Ar, and As (r, s > 0) are isometrically isomorphically equivalent and that the spaces H, Hr, and Hs (r, s > 0) are isometrically isomorphic. These results are then used to reproduce the left-definite spaces and left-definite operators. Furthermore, we will see that our new results imply that the spectra of A and Ar are equal, giving us another proof of this phenomenon that was first established in Littlejohn and Wellman (J Diff Equ 181(2):280–339, 2002).