On the difference of spectral projections

2014/06/25 by Uebersohn, Christoph
#FOS: Mathematics #Functional Analysis (math.FA) #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.1406.6516

Abstract

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_{(-∞, λ)}(T+S) - E_{(-∞, λ)}(T), λ∈ ℝ , of the spectral projections associated with the open interval (-∞, 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 Mart'ınez-Avendaño) on ℓ²(ℕ₀) for all λ∈ ℝ except for at most countably many λ.

Related

• Diagonals of self-adjoint operators II: Non-compact operators
• The spectral density of a difference of spectral projections
• The inverse spectral problem for self-adjoint Hankel operators
• Diagonals of self-adjoint operators I: compact operators
• Diagonals of self-adjoint operators with finite spectrum