Operator-Lipschitz functions in Schatten–von Neumann classes
Tóm tắt
This paper resolves a number of problems in the perturbation theory of linear operators, linked with the 45-year-old conjecure of M. G. Kreĭn. In particular, we prove that every Lipschitz function is operator-Lipschitz in the Schatten–von Neumann ideals S
α
, 1 < α < ∞. Alternatively, for every 1 < α < ∞, there is a constant c
α
> 0 such that
$$ {\left\| {f(a) - f(b)} \right\|_{\alpha }} \leqslant {c_{\alpha }}{\left\| f \right\|_{{{\text{Lip}}\,{1}}}}{\left\| {a - b} \right\|_{\alpha }}, $$
where f is a Lipschitz function with
$$ {\left\| f \right\|_{{{\text{Lip}}\,{1}}}}: = \mathop{{\sup }}\limits_{{_{{\lambda \ne \mu }}^{{\lambda, \mu \in \mathbb{R}}}}} \left| {\frac{{f\left( \lambda \right) - f\left( \mu \right)}}{{\lambda - \mu }}} \right| < \infty, $$
$$ {\left\| \cdot \right\|_{\alpha }} $$
is the norm is S
α
, and a and b are self-adjoint linear operators such that
$$ a - b \in {S^{\alpha }} $$
.
Tài liệu tham khảo
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