  In this short note, we propose to extend differentiability (with respect to a
multidimensional parameter) of a normalized eigenfunction associated to the
simple, dominating eigenvalue of the weighted transfer operator for a uniformly
expanding map, to the whole discrete spectrum. We do so by studying directly
the regularity (with respect to the parameter) of the resolvent operator,
applying a general regularity result for fixed points of maps having loss of
regularity.
