Conditioning Gaussian measure on Hilbert space

For a Gaussian measure on a separable Hilbert space with covariance operator C, we show that the family of conditional measures associated with conditioning on a closed subspace S are Gaussian with covariance operator the short S(C) of the operator C to S. We provide two proofs. The first uses the theory of Gaussian Hilbert spaces and a characterization of the shorted operator by Andersen and Trapp. The second uses recent developments by Corach, Maestripieri and Stojanoff on the relationship between the shorted operator and C-symmetric oblique projections onto S. To obtain the assertion when such projections do not exist, we develop an approximation result for the shorted operator by showing, for any positive operator A, how to construct a sequence of approximating operators A which possess Asymmetric oblique projections onto S such that the sequence of shorted operators S(An) converges to S(A) in the weak operator topology. This result combined with the martingale convergence of random variables associated with the corresponding approximations C establishes the main assertion in general. Moreover, it in turn strengthens the approximation theorem for shorted operator when the operator is trace class; then the sequence of shorted operators S(An) converges to S(A) in trace norm.