Geodesic
Non-nuclear perturbations of discrete operators and trace formulae
Sbornik. Mathematics, Tome 196 (2005) no. 12, pp. 1841-1874

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A trace formula is obtained for unbounded discrete operators perturbed by a Hilbert–Schmidt operator; this formula may be called the discrete analogue of M. Krein's formula for nuclear perturbations. A regularized trace formula of Krein's type is also proved for perturbations in the class $S^p$, $2$, for arbitrary compact and relatively compact perturbations depending on the behaviour at infinity of the distribution function of the spectrum of the unperturbed operator. 
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     author = {Kh. Kh. Murtazin and Z. Yu. Fazullin},
     title = {Non-nuclear perturbations of discrete operators and trace formulae},
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Kh. Kh. Murtazin; Z. Yu. Fazullin. Non-nuclear perturbations of discrete operators and trace formulae. Sbornik. Mathematics, Tome 196 (2005) no. 12, pp. 1841-1874. http://geodesic.mathdoc.fr/item/SM_2005_196_12_a4/