Let E be a separable real Banach space and let Q # L(E*, E) be positive and symmetric. Let S=[S(t)] t 0 be a C 0 -semigroup on E We study the relations between the reproducing kernel Hilbert spaces associated with the operators Q t := t 0 S(s) QS*(s) ds. Under the assumption that these operators are the covariances of centered Gaussian measures + t on E, we also study equivalence + t t+ s for different values of s and t, and we calculate their Radon Nikodym derivatives.
1998 Academic Press
0. Introduction
In this paper we investigate the reproducing kernel Hilbert spaces and Gaussian measures associated with a nonsymmetric Ornstein Uhlenbeck semigroup on a separable real Banach space E. This study is usually carried out in a Hilbert space setting, and one of the motivations of this paper was to see to what extent the theory can be extended to the Banach space setting.
The main difference between the Banach space and the Hilbert space situation is that the covariance operator of a Gaussian measure on a Banach space E is a positive symmetric operator Q (the precise definitions are given in Section 1) from the dual E* into E, rather than an operator on E. Thus, in contrast to the Hilbert space situation, it is no longer possible to represent the reproducing kernel Hilbert space H associated with Q as H=Im Q 1Â2 . When working in a Banach space setting, any reference to the operator Q 1Â2 therefore has to be avoided. This turns out article no. FU973237