Holomorphic functions and the heat kernel measure on an infinite dimensional complex orthogonal group (Q1579896)

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).