 99459 Fritz Gesztesy and Konstantin A. Makarov
 The $\Xi$ Operator and its Relation to Krein's Spectral Shift Function
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Dec 1, 99

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Abstract. We explore connections between Krein's spectral shift function
$\xi(\lambda,H_0,H)$ associated with the pair of selfadjoint operators
$(H_0,H)$, $H=H_0+V$ in a Hilbert space $\calH$ and the recently introduced concept of a spectral shift operator $\Xi(J+K^*(H_0\lambdai0)^{1}K)$
associated with the operatorvalued Herglotz function $J+K^*(H_0z)^{1}K$,
$\Im(z)>0$ in $\calH$, where $V=KJK^*$ and $J=\sgn(V)$. Our principal results include a new representation for $\xi(\lambda,H_0,H)$ in terms of an averaged index for the Fredholm pair of selfadjoint spectral projections
$(E_{J+A(\lambda)+tB(\lambda)}((\infty,0)),E_J((\infty,0)))$, $t\in\bbR$, where $A(\lambda)=\Re(K^*(H_0\lambdai0)^{1}K)$,
$B(\lambda)=\Im(K^*(H_0\lambdai0)^{1}K)$ a.e.~Moreover, introducing the new concept of a trindex for a pair of operators $(A,P)$ in $\calH$, where $A$ is bounded and $P$ is an orthogonal projection, we prove that $\xi(\lambda,H_0,H)$
coincides with the trindex associated with the pair $(\Xi(J+K^*(H_0\lambdai0)^{1}K),\Xi(J))$. In addition, we discuss a
variant of the BirmanKrein formula relating the trindex of a pair of $\Xi$operators and the Fredholm determinant of the abstract scattering matrix.
We also provide a generalization of the classical BirmanSchwinger principle, replacing the traditional eigenvalue counting functions by appropriate spectral shift functions.
This an extended version of a previously archived manuscript.
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