Stochastic integration of operator-valued functions with respect to Banach space-valued Brownian motion

Let E be a real Banach space with property (alpha) and let W-Gamma be an E-valued Brownian motion with distribution Gamma. We show that a function Psi : [0, T] --> L(E) is stochastically integrable with respect to W-Gamma if and only if Gamma-almost all orbits Psi x are stochastically integrable with respect to a real Brownian motion. This result is derived from an abstract result on existence of Gamma-measurable linear extensions of gamma-radonifying operators with values in spaces of gamma-radonifying operators. As an application we obtain a necessary and sufficient condition for solvability of stochastic evolution equations driven by an E-valued Brownian motion.