  In this paper, we consider the linear evolution equation
$dy(t)=Ay(t)dt+Gy(t)dx(t)$, where $A$ is a closed operator, associated to a
semigroup, with good smoothing effects in a Banach space $E$, $x$ is a
nonsmooth path, which is $\eta$-H\"older continuous for some $\eta\in
(1/3,1/2]$, and $G$ is a non-smoothing linear operator on $E$. We prove that
the Cauchy problem associated with the previous equation admits a unique mild
solution and we also show that the solution increases the regularity of the
initial datum as soon as time evolves. Then, we show that the mild solution is
also an integral solution and this allows us to prove a It\^o formula.
