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Functions of perturbed pairs of noncommuting contractions
Izvestiya: Mathematics ( IF 0.8 ) Pub Date : 2020-08-01 , DOI: 10.1070/im8876
A. B. Aleksandrov 1 , V. V. Peller 2, 3
Affiliation  

We consider functions $f(T,R)$ of pairs of noncommuting contractions on Hilbert space and study the problem for which functions $f$ we have Lipschitz type estimates in Schatten--von Neumann norms. We prove that if $f$ belongs to the Besov class $(B_{\infty,1}^1)_+({\Bbb D}^2)$ of analytic functions in the bidisk, then we have a Lipschitz type estimate for functions $f(T,R)$ of pairs of not necessarily commuting contractions $(T,R)$ in the Schatten--von Neumann norms $\boldsymbol{S}_p$ for $p\in[1,2]$. On the other hand, we show that for functions in the Besov space $(B_{\infty,1}^1)_+({\Bbb D}^2)$, there are no Lipschitz such type estimates for $p>2$ as well as in the operator norm.

中文翻译:

非交换收缩的扰动对的函数

我们考虑 Hilbert 空间上非交换收缩对的函数 $f(T,R)$ 并研究我们在 Schatten--von Neumann 范数中具有 Lipschitz 类型估计的函数 $f$ 的问题。我们证明如果 $f$ 属于 Bidisk 中解析函数的 Besov 类 $(B_{\infty,1}^1)_+({\Bbb D}^2)$,那么我们有一个 Lipschitz 类型估计对于 Schatten--von Neumann 范数 $\boldsymbol{S}_p$ for $p\in[1,2] 中的一对不一定交换收缩 $(T,R)$ 的函数 $f(T,R)$ $. 另一方面,我们表明对于 Besov 空间 $(B_{\infty,1}^1)_+({\Bbb D}^2)$ 中的函数,没有 Lipschitz 这样的 $p> 类型估计2$ 以及在运营商规范中。
更新日期:2020-08-01
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