Lipschitz functions of perturbed operators

Abstract

We prove that if $f$ is a Lipschitz function on $\mathbb{R}$, $A$ and $B$ are self-adjoint operators such that $\mathrm{rank}(A-B)=1$, then $f(A)-f(B)$ belongs to the weak space $\boldsymbol{S}_{1,\be}$, i.e., $s_j(A-B)\le \mathrm{const}(1+j{)}^{-1}$. We deduce from this result that if $A-B$ belongs to the trace class ${\mathbf{S}}_1$ and $f$ is Lipschitz, then $f(A)-f(B)\in {\mathbf{S}}_{\Omega }$, 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 $Q=\iint (f(x)-f(y))(x-y{)}^{-1}d{E}_1(x)Td{E}_2(y)$, where $E_1$ and $E_2$ are spectral measures. We show that if $T\in {\mathbf{S}}_1$, then $Q\in {\mathbf{S}}_{\Omega }$ and if $\rank T=1$, then $Q\in\boldsymbol{S}_{1,\be}$. Finally, if $T$ belongs to the Matsaev ideal ${\mathbf{S}}_{\omega }$, then $Q$ is a compact operator.

6 pages