The concept of uniform convexity of a Banach space was
generalized to linear operators between Banach spaces and
studied by Beauzamy. Under this generalization, a Banach space
$X$ is uniformly convex if and only if its identity map $I_X$
is. Pisier showed that uniformly convex Banach spaces have martingale
type $p$ for some $p>1$. We show that this fact is in general not
true for linear operators. To remedy the situation, we
introduce the new concept of martingale subtype and show,
that it is equivalent, also in the operator case, to the
existence of an equivalent uniformly convex norm on $X$.
In the case of identity maps it is also equivalent to
having martingale type $p$ for some $p>1$.
Our main method is to use sequences of ideal norms defined on the
class of all linear operators and to study the factorization of
the finite summation operators. There is a certain analogy with
the theory of Rademacher type.