Asymptotic expansions for trace functionals

We obtain Taylor approximations for functionals V?Tr(f(_H0+V))$V?\mathrm{Tr}(f(H_0+V))$ defined on the bounded self-adjoint operators, where _H0$H_0$ is a self-adjoint operator with compact resolvent and f is a sufficiently nice scalar function, relaxing assumptions on the operators made in 17], and derive estimates and representations for the remainders of these approximations.