The Ξ Operator and Its Relation to Krein’s Spectral Shift Function

We explore connections between Krein’s spectral shift function ξ(λ,H0,H) associated with the pair of self-adjoint operators (H0,H), H = H0 + V in a Hilbert space H and the recently introduced concept of a spectral shift operator Ξ(J +K∗(H0 − λ− i0)−1K) associated with the operatorvalued Herglotz function J + K∗(H0 − z)−1K, Im(z) > 0 in H, where V = KJK and J = sgn(V ). Our principal results include a new representation for ξ(λ,H0,H) in terms of an averaged index for the Fredholm pair of selfadjoint spectral projections (EJ+A(λ)+tB(λ)((−∞, 0)), EJ ((−∞, 0))), t ∈ R, where A(λ) = Re(K∗(H0 − λ − i0)−1K), B(λ) = Im(K∗(H0 − λ − i0)−1K) a.e. Moreover, introducing the new concept of a trindex for a pair of operators (A, P ) in H, where A is bounded and P is an orthogonal projection, we prove that ξ(λ,H0,H) coincides with the trindex associated with the pair (Ξ(J +K∗(H0 − λ− i0)−1K),Ξ(J)). In addition, we discuss a variant of the Birman-Krein formula relating the trindex of a pair of Ξ-operators and the Fredholm determinant of the abstract scattering matrix. We also provide a generalization of the classical Birman-Schwinger principle, replacing the traditional eigenvalue counting functions by appropriate spectral shift functions.