  In this note, we address a question raised by Kr\"otz on the classification
of domains of holomorphy of irreducible admissible Banach representations for
connected non-compact simple real Lie groups G. When G is not of Hermitian
type, we give a complete description of the domains of holomorphy for
irreducible admissible uniformly bounded representations on uniformly convex
uniformly smooth Banach spaces and, in particular, for all irreducible
uniformly bounded Hilbert representations. When the group G is Hermitian, we
determine the domains of holomorphy only when the representations considered
are highest or lowest weight representations.
