A generalized spectral duality theorem
A generalized spectral duality theorem
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广义谱对偶定理
DOI：
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发表时间：
1992
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通讯作者：
W. Chojnacki
中科院分区：
文献类型：
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作者：
W. Chojnacki
我们建立了谱对偶定理的一个版本，将某个协方差代数的*-表示族的点谱与相关的*-表示族的连续谱联系起来。使用该版本，我们通过代数 inl2ℤ 的标准*-表示证明了无理旋转代数的某些*-自同构的某个定点空间的几乎所有元素的图像在这些图像的公共谱的任何非空开子集上都不具有纯点谱。作为谱对偶定理的另一个应用，我们证明，如果几乎所有与 ℝ 上的实数几乎周期函数相关的 Bloch 算子在 ℝ 的 Borel 子集上都具有纯点谱，那么几乎所有具有属于该函数平移的紧壳的势的薛定谔算子在同一集合上都具有纯连续谱。
We establish a version of the spectral duality theorem relating the point spectrum of a family of*-representations of a certain covariance algebra to the continuous spectrum of an associated family of*-representations. Using that version, we prove that almost all the images of any element of a certain space of fixed points of some*-automorphism of an irrational rotation algebra via standard*-representations of the algebra inl2ℤ do not have pure point spectrum over any non-empty open subset of the common spectrum of those images. As another application of the spectral duality theorem, we prove that if almost all the Bloch operators associated with a real almost periodic function on ℝ have pure point spectrum over a Borel subset of ℝ, then almost all the Schrödinger operators with potentials belonging to the compact hull of the translates of this function have, over the same set, purely continuous spectrum.