Spectral shift for relative Schatten class perturbations

We affirmatively settle the question on existence of a real-valued higher order spectral shift function for pair self-adjoint operators $H$ and $V$ such that is bounded $V(H-iI)^{-1}$ belongs to Schatten–von Neumann ideal $\mathcal{S}^n$ compact in separable Hilbert space. also show satisfies same trace formula as known case $V\in\mathcal{S}^n$ it unique up polynomial summand $n-1$. Our result significantly advances earlier partial results where counterparts noncompact perturbations lacked real-valuedness aforementioned uniqueness well appeared more complicated formulas much restrictive sets functions. applies models arising noncommutative geometry mathematical physics.