Gaussian mean boundedness of densely defined linear operators

Abstract

It is known (for details see, Traub, Wasilkowski, and Woźniakowski, 1988, “Information-Based Complexity,” Academic Press, New York) that a linear problem with solution operator S: X → Y in the probabilistic or average case setting has finite ε-complexity with respect to a probability measure μ iff S ϵ L 2( X, μ; Y) or, equivalently, iff I ϵ L 2( Y, μ ° S −1; Y) where I denotes the identity operator and μ ° S −1 is the S-image of μ. If the measure μ is Gaussian and the linear operator S is bounded then μ ° S −1 is also Gaussian and hence I ϵ L p ( Y, μ ° S −1; Y) for any p ≥ 0. We show by two different approaches that this is the case also for linear unbounded densely defined Borel measurable S under the minimal natural condition μ( D( S)) = 1 where D( S) denotes the domain of S. Also we give the expression for the covariance operator of the transformed μ ° S −1.