A distance between operators acting in different Hilbert spaces and operator convergence

The aim of the present article is to give an introduction to the concept of quasi-unitary equivalence and to define several (pseudo-)metrics on the space of self-adjoint operators acting possibly in different Hilbert spaces. As some of the ``metrics'' do not fulfill all properties of a metric (e.g. some lack the triangle inequality or the definiteness), we call them ``distances'' here. To the best of our knowledge, such distances are treated for the first time here. The present article shall serve as a starting point of further research (see e.g. arXiv:2412.13165).