The fundamental role of the Riccati equation for optimal control and optimal filtering of linear systems is well known. For linear distributed parameter systems it has been shown by J. L. Lions [8], and in several other papers, see e.g. [12], [13], [5], that there is an operator Riccati equation (ORE), (or, equivalently, an integro-differential equation of Riccati type for the kernel) associated with various control and filtering problems.