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arXiv · 1605.03931

Estimates on singular values of functions of perturbed operators

Qinbo Liu
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Abstract

This is a conitunation of [1] and [2]. We prove that if function $f$ belongs to the class $Λ_ω \overset{\text{def}}{=} \{f: ω_{f}(δ)\leq \text{const} ω(δ)\} $ for an arbitrary modulus of continuity $ω$, then $s_j(f(A)-f(B))\leq c\cdot ω_{\ast}\big((1+j)^{-\frac{1}{p}}\Vert A-B \Vert_{S_{p}^l}\big) \cdot \Vert f \Vert_{Λ_ω}$ for arbitrary self-adjoint operators $A$, $B$ and all $1\leq j\leq l$, where $ω_{\ast}(x) \overset{\text{def}}{=} x \int_{x}^{\infty}\frac{ω(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.

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Qinbo Liu. 2016-05-17. Estimates on singular values of functions of perturbed operators. https://arxiv.org/abs/1605.03931

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