  In this paper we extend the scope of three important results of the linear
theory of absolutely summing operators. The first one was proved by Bu and
Kranz in \cite{BK} and it asserts that a continuous linear operator between
Banach spaces takes almost unconditionally summable sequences into Cohen
strongly $q$-summable sequences for any $q\geq2$, whenever its adjoint is
$p$-summing for some $p\geq1$. The second of them states that $p$-summing
operators with hilbertian domain are Cohen strongly $q$-summing operators
($1<p,q<\infty$), this result is due to Bu \cite{Bu}. The third one is due to
Kwapie\'{n} \cite{Kwapien} and it characterizes spaces isomorphic to a Hilbert
space using 2-summing operators. We will show that these results are maintained
replacing the hypothesis of the operator to be $p$-summing by almost summing.
We will also give an example of an almost summing operator that fails to be
$p$-summing for every $1\leq p< \infty$.
