Holomorphic functions and the heat kernel measure on an infinite dimensional complex orthogonal group
The author studies Hilbert spaces of holomorphic functions over a particular infinite-dimensional (non-locally compact) noncommutative complex group; viz., the so-called Hilbert-Schmidt orthogonal group \(\text{SO}_{HS}\), embedded in the space of Hilbert-Schmidt operators on an infinite-dimensional Hilbert space. The class \(\text{so}_{HS}\) of skew symmetric Hilbert-Schmidt operators plays the role of a Lie algebra of \(\text{SO}_{HS}\). Let \(\text{SO}(\infty)\) denote the closure of the direct limit of a certain sequence of finite-dimensional subgroups of \(\text{SO}_{HS}\) in a Riemannian metric. To define the Hilbert spaces \({\mathcal H}L^2(\text{SO}_{HS}, \mu_t)\) and \({\mathcal H}L^2(\text{SO}(\infty))\) of holomorphic functions, the author first constructs the heat kernel measure \(\mu_t\) on \(\text{SO}_{HS}\). In the finite-dimensional case this measure has a density with respect to Haar measure, making it possible to apply techniques from partial differential equations. In the present infinite-dimensional case, however, stochastic differential equations in infinite dimensions are used to define \(\mu_t\) as the transition probability of a diffusion on \(\text{SO}_{HS}\). All holomorphic polynomials (in the matrix entries) on \(\text{SO}_{HS}\) are square integrable with respect to the heat kernel measure, and \({\mathcal H}L^2(\text{SO}_{HS}, \mu_t)\) is the closure in \(L^2(\text{SO}_{HS}, \mu_t)\) of the space of the holomorphic polynomials. \({\mathcal H}L^2(\text{SO}(\infty))\) denotes a certain Hilbert space of holomorphic functions on \(\text{SO}(\infty)\) with a direct limit type norm. It is proved that the embedding of the space of holomorphic polynomials into \({\mathcal H}L^2(\text{SO}(\infty))\) can be extended to an isometry from \({\mathcal H}L^2(\text{SO}_{HS}, \mu_t)\) into \({\mathcal H}L^2(\text{SO}(\infty))\). The main theorem is that an infinite-dimensional nonlinear analog of the Taylor expansion defines an isometry from \({\mathcal H}L^2(\text{SO}(\infty))\) into the Hilbert space \(J^0_t\) associated with a Lie subalgebra of \(\text{so}_{HS}\). This is an extension to infinite dimensions of an isometry of \textit{B. Driver} and \textit{L. Gross} for complex Lie groups [cf. ``New trends in stochastic analysis'', Charingworth, 1994, 76-106, World Scientific Publ. (1997; Zbl 0944.60012)]; \(\text{SO}(\infty)\) plays the role of a Cameron-Martin subspace for the group. The author notes that her methods can be applied in more general situations than in the case of one concrete group (as outlined here).
holomorphic functions on an infinite-dimensional group
scientific article; zbMATH DE number 1507482