We are going to consider the general problem whether the sum of
two closed operators on a Banach space is closed on the
intersection of their domains. We introduce absolute functional
calculus for sectorial operators, which is stronger than
$H^\infty$-calculus. Using this technique, we prove a theorem of
Dore-Venni type for sums of closed operators. There, we are able
to remove any assumptions such as R-boundedness or BIP on one of
the operators given that the second operator has absolute
calculus. Moreover, we show that any sectorial operator has
absolute calculus on the real interpolation spaces between its
domain and the space itself.