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, $\mathcal{ B}_p(\mathcal{ H})$, $p \in [1,\infty)$.
Categories: math.FA math-ph math.MP math.SP