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24 June 2024
 
  » arxiv » 2302.00218

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An integral formula for the projection constant of the trace class
Andreas Defant ; Daniel Galicer ; Martín Mansilla ; Mieczysław Mastyło ; Santiago Muro ;
Date 1 Feb 2023
AbstractWe study the projection constant $oldsymbol{lambda}ig(mathcal S_1(n)ig)$ of the trace class $mathcal S_1(n)$ of all operators on the linebreak $n$-dimensional complex Hilbert space, and prove the integral formula $oldsymbol{lambda}ig(mathcal S_1(n)ig) = n int_{mathcal U_n} vert ext{tr}(U) vert ,dU ,,$ where the integration is with respect to the Haar measure on the group $mathcal U_n$ of unitary operators. The approach we use is based on harmonic polynomials on the unitary group. Using probabilistic methods, we show that $ oldsymbol{lambda}ig(mathcal S_1(n)ig)/n o sqrt{pi}/2 $ as $n o infty$,.
Source arXiv, 2302.00218
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