The projection constant for the trace class

Autor:	Defant, Andreas, Galicer, Daniel, Mansilla, Martín, Mastyło, Mieczysław, Muro, Santiago
Rok vydání:	2023
Předmět:	Mathematics - Functional Analysis Mathematics - Metric Geometry 46B06 46B07 46G25 (Primary) 06E30 30B50 47B10 43A15 (Secondary)
Druh dokumentu:	Working Paper
Popis:	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 $ \boldsymbol{\lambda}\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} \boldsymbol{\lambda}\big(\mathcal S_1(n)\big)/n = \sqrt{\pi}/2\,. $ Comment: arXiv admin note: substantial text overlap with arXiv:2208.06467
Databáze:	arXiv
Externí odkaz:	http://arxiv.org/abs/2302.00218 Zobrazit plný text záznamu View this record from Arxiv