Unitary approximation of positive operators

Abstract

Of concern are some operators inequalities arising in quantum chemistry. Let $A$ be a positive operator on a Hilbert space $\mathcal{H}$. We consider the minimization of $||U-A||_{p}$ as $U$ ranges over the unitary operators in $\mathcal{H}$ and prove that in most cases the minimum is attained when $U$ is the identity operator. The norms considered are the Schatten $p$-norms. The methods used are of independent interest; application is made of noncommutative differential calculus.