  In this paper we extend dyadic shifts and the dyadic representation theorem
to an operator-valued setting: We first define operator-valued dyadic shifts
and prove that they are bounded. We then extend the dyadic representation
theorem, which states that every scalar-valued Calder\'on-Zygmund operator can
be represented as a series of dyadic shifts and paraproducts averaged over
randomized dyadic systems, to operator-valued Calder\'on-Zygmund operators. As
a corollary, we obtain another proof of the operator-valued, global T(1)
theorem.
  We work in the setting of integral operators that have R-bounded
operator-valued kernels and act on functions taking values in UMD-spaces. The
domain of the functions is the Euclidean space equipped with the Lebesgue
measure.
  In addition, we give new proofs for the following known theorems: Boundedness
of the dyadic (operator-valued) paraproduct, a variant of Pythagoras' theorem
for (vector-valued) functions adapted to a sparse collection of dyadic cubes,
and a decoupling inequality for (UMD-valued) martingale differences.
