Conditioning Gaussian measure on Hilbert space

Abstract

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^n which possess A^n- symmetric oblique projections onto S^⊥ such that the sequence of shorted operators S(A^n) 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^n 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(A^n) converges to S(A) in trace norm.