For anyone who's familiar with G. Da Prato's books on infinite dimensional analysis, I was wondering if someone could clarify something. In, for instance, "An Introduction to Infinite Dimensional Analysis" and "Second Order Partial Differential Equations in Hilbert Spaces", Da Prato introduces symmetric operators on Hilbert spaces as covariance operators for Gaussians. Since these operators are bounded, they are self-adjoint. In other places, he just mentions symmetric operators, without making it clear if he means them to be bounded. I am wondering, does Da Prato mean for his symmetric operators to all be self-adjoint?