2015/06/13 by Houman Owhadi, Owhadi, Houman, Clint Scovel +1
Mathematics · #28C20 #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Mathematical Approximation and Integration #Numerical methods in inverse problems #Probability (math.PR) #math.PR #msc:28C20
paper · pdf · doi:10.48550/arxiv.1506.04208
openalex publication_date 2015/06/13 · arxiv created 2015/09/01 · arxiv updated 2015/09/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 An which possess An-symmetric 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 Cn 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.