Some applications of almost analytic extensions to operator bounds in trace ideals

Abstract

Using the Davies-Helffer-Sjöstrand 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 $\Vert f(S_2)-f(S_1){\Vert }_{\mathcal{B}(\mathcal{H})}$ in terms of the norm of the difference of resolvents, $\Vert (S_2-z_0{I}_{\mathcal{H}}{)}^{-1}-(S_2-z_0{I}_{\mathcal{H}}{)}^{-1}{\Vert }_{\mathcal{B}(\mathcal{H})}$, 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, ${\mathcal{B}}_p(\mathcal{H})$, $p\in [1,\infty )$.

20 pages, references and Remark 3.3 got updated

References in corpus (1)

• On the Witten index in terms of spectral shift functions

Cited by in corpus (1)

• Functions of self-adjoint operators in ideals of compact operators