Estimates on singular values of functions of perturbed operators

TL;DR

This paper provides bounds on the singular values of functions of perturbed operators, extending previous results to a broad class of operators and functions characterized by a modulus of continuity.

Contribution

It generalizes estimates on singular values of functions of operators to include various classes of operators and functions with arbitrary moduli of continuity.

Findings

Derived bounds for singular values of functions of self-adjoint operators.

Extended results to contractions, dissipative, normal operators, and commuting tuples.

Provided a unified approach for different operator classes and continuity moduli.

Abstract

This is a conitunation of [1] and [2]. We prove that if function $f$ belongs to the class $\Lambda_{\omega} \overset{\text{def}}{=} \{f: \omega_{f}(\delta)\leq \text{const} \omega(\delta)\} $ for an arbitrary modulus of continuity $\omega$, then $s_j(f(A)-f(B))\leq c\cdot \omega_{\ast}\big((1+j)^{-\frac{1}{p}}\Vert A-B \Vert_{S_{p}^l}\big) \cdot \Vert f \Vert_{\Lambda_{\omega}}$ for arbitrary self-adjoint operators $A$, $B$ and all $1\leq j\leq l$, where $\omega_{\ast}(x) \overset{\text{def}}{=} x \int_{x}^{\infty}\frac{\omega(t)}{t^2}dt ( x>0) $. The result is then generalized for contractions, maximal dissipative operators, normal operators and $n$-tuples of commuting self-adjoint operators.