  In this paper we define a new type of quadratic variation for cylindrical
continuous local martingales on an infinite dimensional spaces. It is shown
that a large class of cylindrical continuous local martingales has such a
quadratic variation. For this new class of cylindrical continuous local
martingales we develop a stochastic integration theory for operator valued
processes under the condition that the range space is a UMD Banach space. We
obtain two-sided estimates for the stochastic integral in terms of the
$\gamma$-norm. In the scalar or Hilbert case this reduces to the
Burkholder-Davis-Gundy inequalities. An application to a class of stochastic
evolution equations is given at the end of the paper.
