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

TL;DR

This paper explores how almost analytic extensions can be used to estimate operator bounds in trace ideals, focusing on functions of self-adjoint operators and their resolvent differences.

Contribution

It introduces a method using almost analytic extensions to control operator norms in trace ideals based on resolvent differences.

Findings

Established bounds for operator differences in trace ideals.

Extended functional calculus techniques to trace ideal settings.

Provided new estimates for functions of self-adjoint operators.

Abstract

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, $\mathcal{B}_p(\mathcal{H})$, $p \in [1,\infty)$.