The Ξ operator and its relation to Krein's spectral shift function

Abstract

We explore connections between Krein's spectral shift function ζ(λ,H 0, H) associated with the pair of self-adjoint operators (H 0, H),H=H 0+V, in a Hilbert spaceH and the recently introduced concept of a spectral shift operator Ξ(J+K *(H 0−λ−i0)−1 K) associated with the operator-valued Herglotz functionJ+K *(H 0−z)−1 K, Im(z)>0 inH, whereV=KJK * andJ=sgn(V). Our principal results include a new representation for ζ(λ,H 0,H) in terms of an averaged index for the Fredholm pair of self-adjoint spectral projections (E J+A(λ)+tB(λ)(−∞, 0)),E J((−∞, 0))), ℝ, whereA(λ)=Re(K *(H 0−λ−i0−1 K),B(λ)=Im(K *(H 0−λ-i0)−1 K) a.e. Moreover, introducing the new concept of a trindex for a pair of operators (A, P) inH, whereA is bounded andP is an orthogonal projection, we prove that ζ(λ,H 0, H) coincides with the trindex associated with the pair (Ξ(J+K *(H 0−λ−i0)K), 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.