arXiv subjects
Publications and source records attributed to Christoph Uebersohn.
We generalise a classical example given by Krein in 1953. We compute the difference of the resolvents and the difference of the spectral projections explicitly. We further give a full description of the unitary invariants, i.e., of the spectrum and the multiplicity. Moreover, we observe a link between the difference of the spectral projections and Hankel operators.
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$.
In this paper we perform an explicit diagonalization of Hankel integral operators $ K^{(0)}, K^{(1)}, K^{(2)}, ... $ It turns out that each of these operators has a simple purely absolutely continuous spectrum filling in the interval $ [-1,1] $. This generalizes a result of Kostrykin and Makarov (2008).