  Using the Davies-Helffer-Sj\"ostrand functional calculus based on almost
analytic extensions, we address the following problem: Given self-adjoint
operators $S_j$, $j=1,2$, in $\mathcal{H}$, and functions $f$ in an appropriate
class, for instance, $f \in C_0^{\infty}(\mathbb{R})$, how to control the norm
$\|f(S_2) - f(S_1)\|_{\mathcal{B}(\mathcal{H})}$ in terms of the norm of the
difference of resolvents, $\|(S_2 - z_0 I_{\mathcal{H}})^{-1} - (S_2 - z_0
I_{\mathcal{H}})^{-1}\|_{\mathcal{B}(\mathcal{H})}$, for some $z_0 \in
\mathbb{C}\backslash\mathbb{R}$. We are particularly interested in the case
where $\mathcal{B}(\mathcal{H})$ is replaced by a trace ideal,
$\mathcal{B}_p(\mathcal{H})$, $p \in [1,\infty)$.
