  Let $H$ be the Hardy operator and $I$ the identity operator acting on
functions on the real half-line. We find optimal bounds for the operator $H -
I$ in the setting of power weights and the cases of positive decreasing
functions, positive functions, and general functions. As a byproduct, we obtain
some results about the optimal relations between the norms of $H$ and its dual.

Classical Analysis and ODEs;Functional Analysis