In a Banach algebra, do ab and ba have almost the same exponential spectrum?

Bonjour Yemon, Oui, c'est un problème proposé par TJR dans le cadre d'un projet de recherche d'été CRSNG.

It is indeed a very subtle question, I thought it might interest some people here. This problem appears to be strongly related to the topology of the group of invertible elements, which is difficult to study.

And yes, one can show that the exponential spectrum of a*b is the same than the one of b*a in the Calkin algebra. It follows from the fact that 1-ab is of Fredholm index zero if and only if 1-ba is of Fredholm index zero.

Just to update this: a negative solution was recently given by Klaja and Ransford. See arXiv 1510.08109.