Energy conservation and unitary approximation numbers

The role of unitary operators is central in mathematics, in questions related to energy conservation in physics, as well as in stable computations in numerical linear algebra and scientific simulations. In this paper a refined way to measure the deviation from unitary operators is proposed for matrices in terms of so-called unitary approximation numbers. These describe how far an operator is from being unitary, i.e., energy conserving modulo small rank perturbations.