Gaussian mean boundedness of densely defined linear operators

It is known (for details see, Traub, Wasilkowski, and Wofniakowski, 1988, “Information-Based Complexity,” Academic Press, New York) that a linear problem with solution operator S: X --f Y in the probabilistic or average case setting has finite e-complexity with respect to a probability measure k iff S E L2(X, CL; Y) or, equivalently, iff I E L2( Y, h 0 S-i; Y) where I denotes the identity operator and p 0 S-i is the S-image of p. If the measure p is Gaussian and the linear operator S is bounded then p 0 S-’ is also Gaussian and hence I E Lp( Y, p 0 S-i; Y) for any p 2 0. We show by two different approaches that this is the case also for linear unbounded densely defined Bore1 measurable S under the minimal natural condition p@(S)) = 1 where D(S) denotes the domain of S. Also we give the expression for the covariance operator of the transformed p 0 S-l. o 1~ Academic Ress, Inc.