The projection constant for the trace class
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arXiv
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| Hauptverfasser: | , , , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866917918022828032 |
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| author | Defant, Andreas Galicer, Daniel Mansilla, Martín Mastyło, Mieczysław Muro, Santiago |
| author_facet | Defant, Andreas Galicer, Daniel Mansilla, Martín Mastyło, Mieczysław Muro, Santiago |
| contents | We study the projection constant of the space of operators on $n$-dimensional Hilbert spaces, with the trace norm, $\mathcal S_1(n)$. We show an integral formula for the projection constant of $\mathcal S_1(n)$; namely $ \boldsymbolλ\big(\mathcal S_1(n)\big) = n \int_{\mathcal U_n} \vert \text{tr}(U) \vert \,dU \,, $ where the integration is with respect to the Haar probability measure on the group $\mathcal U_n$ of unitary operators. Using a probabilistic approach, we derive the limit formula $ \lim_{n\to \infty} \boldsymbolλ\big(\mathcal S_1(n)\big)/n = \sqrtπ/2\,. $ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2302_00218 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The projection constant for the trace class Defant, Andreas Galicer, Daniel Mansilla, Martín Mastyło, Mieczysław Muro, Santiago Functional Analysis Metric Geometry 46B06, 46B07, 46G25 (Primary) 06E30, 30B50, 47B10, 43A15 (Secondary) We study the projection constant of the space of operators on $n$-dimensional Hilbert spaces, with the trace norm, $\mathcal S_1(n)$. We show an integral formula for the projection constant of $\mathcal S_1(n)$; namely $ \boldsymbolλ\big(\mathcal S_1(n)\big) = n \int_{\mathcal U_n} \vert \text{tr}(U) \vert \,dU \,, $ where the integration is with respect to the Haar probability measure on the group $\mathcal U_n$ of unitary operators. Using a probabilistic approach, we derive the limit formula $ \lim_{n\to \infty} \boldsymbolλ\big(\mathcal S_1(n)\big)/n = \sqrtπ/2\,. $ |
| title | The projection constant for the trace class |
| topic | Functional Analysis Metric Geometry 46B06, 46B07, 46G25 (Primary) 06E30, 30B50, 47B10, 43A15 (Secondary) |
| url | https://arxiv.org/abs/2302.00218 |