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arXiv 1406.6516math.FAmath.SP
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关于谱投影的差分

On the difference of spectral projections

Christoph Uebersohn

备注：
23 pages, 1 figure. This paper generalizes the results from arXiv:1406.6516v1
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摘要：

对于作用在无穷维复可分Hilbert空间上的半有界自伴算子$ T $和紧自伴算子$ S $，研究了开区间$ (-\infty, λ) $上谱投影的差$ D(λ) := E_{(-\infty, λ)}(T+S) - E_{(-\infty, λ)}(T), \, λ\in \mathbb{R} $.在$ S $是秩1的情况下，我们证明了$ D(λ) $是酉等价于块对角算子$ Γ_λ \oplus 0 $，其中$ Γ_λ $是有界自伴汉克尔算子，对于所有的$ λ\in \mathbb{R} $，除了至多可数个$ λ$。更一般地，如果$ S $是紧的，那么我们得到$ D(λ) $酉等价于$ \ell^{2}(\mathbb{N}_{0}) $上的本质Hankel算子（在Mart\'ınez-Avendaño的意义下），对于除了至多可数个$ λ$之外的所有$ λ\in \mathbb{R} $。

英文摘要：

For a semibounded self-adjoint operator $ T $ and a compact self-adjoint operator $ S $ acting on a complex separable Hilbert space of infinite dimension, we study the difference $ D(λ) := E_{(-\infty, λ)}(T+S) - E_{(-\infty, λ)}(T), \, λ\in \mathbb{R} $, of the spectral projections associated with the open interval $ (-\infty, λ) $. In the case when $ S $ is of rank one, we show that $ D(λ) $ is unitarily equivalent to a block diagonal operator $ Γ_λ \oplus 0 $, where $ Γ_λ $ is a bounded self-adjoint Hankel operator, for all $ λ\in \mathbb{R} $ except for at most countably many $ λ$. If, more generally, $ S $ is compact, then we obtain that $ D(λ) $ is unitarily equivalent to an essentially Hankel operator (in the sense of Mart\'ınez-Avendaño) on $ \ell^{2}(\mathbb{N}_{0}) $ for all $ λ\in \mathbb{R} $ except for at most countably many $ λ$.

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