Heat Kernel Analysis and Cameron–Martin Subgroup for Infinite Dimensional Groups

The heat kernel measure μt 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 GCM is defined and its properties are discussed. In particular, there is an isometry from the L2μt-closure of holomorphic polynomials into a space Ht(GCM) of functions holomorphic on GCM. This means that any element from this L2μt-closure of holomorphic polynomials has a version holomorphic on GCM. In addition, there is an isometry from Ht(GCM) 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.