Lipschitz Estimates and an application to trace formulae
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866918130719129600 |
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| author | Bhattacharyya, Tirthankar Chattopadhyay, Arup Giri, Saikat Pradhan, Chandan |
| author_facet | Bhattacharyya, Tirthankar Chattopadhyay, Arup Giri, Saikat Pradhan, Chandan |
| contents | In this note, we provide an elementary proof for the expression of $f(U)-f(V)$ in the form of a double operator integral for every Lipschitz function $f$ on the unit circle $\cir$ and for a pair of unitary operators $(U,V)$ with $U-V\in\mathcal{S}_{2}(\hilh)$ (the Hilbert-Schmidt class). As a consequence, we obtain the Schatten $2$-Lipschitz estimate $\|f(U)-f(V)\|_2\leq \|f\|_{\lip(\cir)}\|U-V\|_2$ for all Lipschitz functions $f:\cir\to\C$. Moreover, we develop an approach to the operator Lipschitz estimate for a pair of contractions with the assumption that one of them is a strict contraction, which significantly extends the class of functions from results known earlier. More specifically, for each $p\in(1,\infty)$ and for every pair of contractions $(T_0,T_1)$ with $\|T_0\|<1$, there exists a constant $d_{f, p,T_0}>0$ such that $\|f(T_1)-f(T_0)\|_p\leq d_{f,p, T_0}\|T_1-T_0\|_p$ for all Lipschitz functions on $\cir$. Using our Lipschitz estimates, we establish a modified Krein trace formula applicable to a specific category of pairs of contractions featuring Hilbert-Schmidt perturbations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_08706 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Lipschitz Estimates and an application to trace formulae Bhattacharyya, Tirthankar Chattopadhyay, Arup Giri, Saikat Pradhan, Chandan Functional Analysis 47A20, 47A55, 47A56, 47B10 In this note, we provide an elementary proof for the expression of $f(U)-f(V)$ in the form of a double operator integral for every Lipschitz function $f$ on the unit circle $\cir$ and for a pair of unitary operators $(U,V)$ with $U-V\in\mathcal{S}_{2}(\hilh)$ (the Hilbert-Schmidt class). As a consequence, we obtain the Schatten $2$-Lipschitz estimate $\|f(U)-f(V)\|_2\leq \|f\|_{\lip(\cir)}\|U-V\|_2$ for all Lipschitz functions $f:\cir\to\C$. Moreover, we develop an approach to the operator Lipschitz estimate for a pair of contractions with the assumption that one of them is a strict contraction, which significantly extends the class of functions from results known earlier. More specifically, for each $p\in(1,\infty)$ and for every pair of contractions $(T_0,T_1)$ with $\|T_0\|<1$, there exists a constant $d_{f, p,T_0}>0$ such that $\|f(T_1)-f(T_0)\|_p\leq d_{f,p, T_0}\|T_1-T_0\|_p$ for all Lipschitz functions on $\cir$. Using our Lipschitz estimates, we establish a modified Krein trace formula applicable to a specific category of pairs of contractions featuring Hilbert-Schmidt perturbations. |
| title | Lipschitz Estimates and an application to trace formulae |
| topic | Functional Analysis 47A20, 47A55, 47A56, 47B10 |
| url | https://arxiv.org/abs/2312.08706 |