Estimates on singular values of functions of perturbed operators

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.