Heat kernel analysis and Cameron-Martin subgroup for infinite dimensional groups

The heat kernel measure mu(t) is constructed on GL(H), the group of invertible operators on a complex Hilbert space H. This measure is determined by an infinite dimensional Lie algebra g and a Hermitian inner product on it. The Cameron-Martin subgroup G(CM) is defined and its properties are discussed. In particular, there is an isometry from the L-mu i(2)-closure of holomorphic polynomials into a space H-t(G(CM)) of functions holomorphic on G(CM). This means that any element from this L-mu t(2)-closure of holomphoric polynomials has a version holomphoric on G(CM). In addition, there is an isometry from H-t(G(CM)) into a Hilbert space associated with the tensor algebra over g. The latter isometry is an infinite dimensional analog of the Taylor expansion. As examples we discuss a complex orthogonal group and a complex symplectic group. (C) 2000 Academic Press.