Lipschitz functions of perturbed operators

TL;DR

This paper investigates how Lipschitz functions affect perturbed operators, showing that small rank or trace class perturbations lead to specific spectral behavior and operator class inclusions, with implications for double operator integrals.

Contribution

It establishes new bounds on the spectral properties of Lipschitz functions of operators under rank-one and trace class perturbations, extending understanding of operator differences.

Findings

If A-B has rank 1, then f(A)-f(B) is in the weak Schatten class S_{1,∞}.

If A-B is trace class, then f(A)-f(B) belongs to the logarithmic Schatten class S_Ω.

Double operator integrals with T in S_1 are in S_Ω; rank-one T yields S_{1,∞}; T in S_ω implies compactness.

Abstract

We prove that if $f$ is a Lipschitz function on $\R$, $A$ and $B$ are self-adjoint operators such that ${\rm rank} (A-B)=1$, then $f(A)-f(B)$ belongs to the weak space $\boldsymbol{S}_{1,\be}$, i.e., $s_j(A-B)\le{\rm const} (1+j)^{-1}$. We deduce from this result that if $A-B$ belongs to the trace class $\boldsymbol{S}_1$ and $f$ is Lipschitz, then $f(A)-f(B)\in\boldsymbol{S}_\Omega$, i.e., $\sum_{j=0}^ns_j(f(A)-f(B))\le\const\log(2+n)$. We also obtain more general results about the behavior of double operator integrals of the form $Q=\iint(f(x)-f(y))(x-y)^{-1}dE_1(x)TdE_2(y)$, where $E_1$ and $E_2$ are spectral measures. We show that if $T\in\boldsymbol{S}_1$, then $Q\in\boldsymbol{S}_\Omega$ and if $\rank T=1$, then $Q\in\boldsymbol{S}_{1,\be}$. Finally, if $T$ belongs to the Matsaev ideal $\boldsymbol{S}_\omega$, then $Q$ is a compact operator.