On an Integral Formula for Fredholm Determinants Related to Pairs of Spectral Projections
Integral Equations and Operator Theory
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We consider Fredholm determinants of the form identity minus product of spectral projections corresponding to isolated parts of the spectrum of a pair of self-adjoint operators. We show an identity relating such determinants to an integral over the spectral shift function in the case of a rank-one perturbation. More precisely, we prove -ln(det(1-1I(A)1R\I(B)1I(A)))=∫Idx∫R\Idyξ(x)ξ(y)(y-x)2,where 1 J (·) denotes the spectral projection of a self-adjoint operator on a set J∈ Borel (R). The operators A and B are self-adjoint, bounded from below and differ by a rank-one perturbation and ξ denotes the corresponding spectral shift function. The set I is a union of intervals on the real line such that its boundary lies in the resolvent set of A and B and such that the spectral shift function vanishes there i.e. I contains isolated parts of the spectrum of A and B. We apply this formula to the subspace perturbation problem.
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