The geometry of the projective joint spectrum and the commutativity of self-adjoint operators

The theory of single operators is by now a very mature subject, with the notion of spectrum playing a key role in the theory. However, multivariate operator theory is only in its very early stages of development. There is not even wide agreement about how "the joint spectrum" of an n-tuple A = (A 1, ˙ ˙ ˙ , An) of bounded linear operators on the same Hilbert space H should be defined. In this thesis, we investigate the geometry of the projective joint spectrum defined by Yang. We fully characterize, in certain settings, the commutativity of the n-tuple of bounded linear operators A on a separable Hilbert space H, in terms of the structure of the proper part of the projective joint spectrum.