Singular traces and perturbation formulae of higher order

Let $H,V$ be self-adjoint operators such that $V$ belongs to the weak trace class ideal. We prove higher order perturbation formula $$\tau (f(H+V)-\sum _{j=0}^{n-1}\frac{1}{j!}\frac{d^j}{d{t}^j}f(H+tV){\mid }_{t=0})={\int }_{\mathbb{R}}{f}^{(n)}(t)d{m}_n(t),$$ where $\tau$ is a trace on the weak trace class ideal and $m_n$ is a finite measure that is not necessarily absolutely continuous. This result extends the first and second order perturbation formulas of Dykema and Shripka, who generalised the Krein and Koplienko trace formulas to the weak trace class ideal. We also establish the perturbation formulae when the perturbation $V$ belongs to the quasi-Banach ideal weak-$L_n$ for any $n\ge 1$.