Groupoid Techniques in Operator Theory

The spectral multiplicity theory is a classical topic in operator theory, whose aim is to describe the unitary orbits of normal operators on Hilbert space, which are just orbits of the similarity action of the unitary group on the set of normal operators. We show that, in the case of infinite-dimensional Hilbert spaces, the study of norm closures of the unitary orbits is most naturally performed in the framework of groupoids and groupoid actions.