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0 Let us now come to the easier term IIδ . We write as before IIδ = ∞ R2d ∞ 0 + 0 ∞ + 0 R R Fδ (t, H0 (X )) [ A δ (X ) − A (X )] ζ (t) ϕ(H0 (X )) dt d X h 0 (E) [F(t, E) − Fδ (t, E)] ( h 0 (E) [F(t, E) − Fδ (t, E)] [( A )δ (E) ζ (t) ϕ(H0 (X )) dt d E A )(E) − ( A )δ (E)] ×ζ (t) ϕ(E) dt d E, where we defined the regularization of ( ( A )δ (E) := R ( A )(E) in the energy variable E, A )(E − E ) χδ s (E ) d E = R ( A )(E −δ s E ) χ (E ) d E , (14) and s ∈]0, 1/2[ is a parameter that can be chosen arbitrarily.

If D = D ∗ ηN and Rz (D) ∈ K(N , τ ) for some (and hence for all) z ∈ C \ R, then we say that D has τ -compact resolvent. An operator T ∈ N is called τ -Fredholm [29, Appendix B] if the projections [ker T ], [ker T ∗ ] are τ -finite and there exists a τ -finite projection E ∈ N such that ran(1 − E) ⊆ ran(T ). For a τ -Fredholm operator T one can define its τ -index by τ - ind(T ) = τ ([ker T ]) − τ ([ker T ∗ ]). 1. Let N be a semifinite von Neumann algebra and let E, E be two invariant operator ideals over N .

15, Appendix B, Lemma 6] If D0 is an unbounded self-adjoint operator, A is a bounded self-adjoint operator, and D = D0 + A then (1 + D 2 )−1 √ where f (a) = 1 + 21 a 2 + 21 a a 2 + 4. 7. Let D0 = D0∗ ηN have τ -compact resolvent, and let B R = {V = V ∗ ∈ N : V R} . Then for any compact subset ⊆ R the function V ∈ B R → E D0 +V is L1 (N , τ )-bounded. Proof. We have E D0 +V c0 (1 + (D0 + V )2 )−1 for some constant c0 = c0 ( ) > 0 and for every V = V ∗ ∈ N . 6 there exists a constant c1 = c1 (R) > 0, such that for all V ∈ B R , (1 + (D0 + V )2 )−1 c1 (1 + D02 )−1 .