On the Difference of Spectral Projections

Abstract

For a semibounded self-adjoint operator T and a compact selfadjoint operator S acting on a complex separable Hilbert space of infinite dimension, we study the difference D(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 ⊕ 0, where Γ λ is a bounded self-adjoint Hankel operator, for all λ ∈ 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 Martínez-Avendaño) on ℓ 2 (N0) for all λ ∈ R except for at most countably many λ.

On a generalisation of Krein's example

“…Relations between differences of spectral projections and Hankel operators are also discussed in the work of Pushnitski [22,23,24] and together with Yafaev [25,26] in the framework of scattering theory, related to an idea of Peller [18]. We also refer to [28] for an approach based on a result of Megretskiȋ, Peller, and Treil [16].…”