Let $T(t),0\leq t<\infty ,$ be a one parameter $C$-semigroup 
of bounded linear operators on a Banach space $X,$ and $A$ 
be the generator of $T(t)$.
Let $R(\lambda ,A)$ be the resolvent operator of $A$. It is 
known that for exponentially bounded $C$-semigroups, 
$\left\| R(\lambda ,A)C\right\| \leq \frac M{\lambda -\omega }$ 
$\;$for $\lambda >\omega .$ The object of this paper is to 
study such an inequality for the $\left( p,q\right) $-summing
norms. Further, we give some conditions for a $C$-semigroup 
to be in the ideal of $\left( p,q\right) $-summing operators.