The projection constant for the trace class

TL;DR

This paper derives an integral formula for the projection constant of the trace class operators on n-dimensional Hilbert spaces and establishes its asymptotic behavior as the dimension grows large.

Contribution

It provides a new integral formula for the projection constant of the trace class and analyzes its limit using probabilistic methods.

Findings

Integral formula for the projection constant involving Haar measure

Asymptotic limit of the normalized projection constant as n approaches infinity

Probabilistic approach to derive the limit formula

Abstract

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\,. $