Spectral theory is a subdiscipline of Functional Analysis which uses generalised eigenvalues of bounded linear operators to describe phenomena in infinite-dimensional spaces. The correct setting for this theory is the concept of a Banach algebra which extends the notion of an algebra of matrices. In this lecture, which will be accessible to a general audience, Dr. Schulz will explore when the similarity orbits of two Banach algebra elements commute with each other which, amongst others, leads to generalisations of well-known classical results by Zemanek.