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).