Operator integration, perturbations and commutators

Under mild conditions integral representations of the following kind are justified:
$$ f(A_1)\cdot J-J\cdot f(A_0)=\iint\frac{f(\mu)-f(\lambda)}{\mu-\lambda}d \,E_1(\mu)(A_1J-JA_0)d \,E_0(\mu). \qquad{(*)} $$
Here $A_k$, $k=0,1$, is a self-adjoint operator on a Hilbert space $\mathcal{H}_k$, $J$ is an operator acting from $\mathcal{H}_0$ to $\mathcal{H}_1$; all operators are, in general, unbounded; $E_k$ is the spectral measure for $A_k$. On the basis of the representation ($*$) estimates of s-numbers of the operator $f(A_1)\cdot J-J\cdot f(A_0)$ in terms of the $s$-numbers of $A_1J-JA_0$ are given. Analogous results are obtained for commutators and anticommutators.