Linear estimators and radonifying operators

We consider the problem of estimating a signal $Y$ with values in a Banach space based on the observation $X$ with values in another Banach space given their joint Gaussian distribution. Linear estimators are denned to be measurable linear transformations. A characterization of measurable linear transformations with respect to a Gaussian measure by radonifying operators is established. The Bayes estimator $\mathbf{E}(Y|X)$ is shown to be a measurable linear transformation and the associated radonifying operator is derived.