Fréchet interpolation spaces and Grothendieck operator ideals

Starting with a continuous injection $I:X\longrightarrow Y$ between Banach spaces, we are interested in the Fréchet (non Banach) space obtained as the reduced projective limit of the real interpolation spaces. We study relationships among the pertenence of $I$ to an operator ideal and the pertenence of the given interpolation space to the Grothedieck class generated by that ideal