Heat Kernel Analysis and Cameron–Martin Subgroup for Infinite Dimensional Groups
✍ Scribed by Maria Gordina
- Publisher
- Elsevier Science
- Year
- 2000
- Tongue
- English
- Weight
- 367 KB
- Volume
- 171
- Category
- Article
- ISSN
- 0022-1236
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✦ Synopsis
The heat kernel measure + 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 2 +t -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 2 +t -closure of holomorphic polynomials has a version holomorphic 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.