Heat kernel analysis of Cameron-Martin subgroup for infinite dimensional groups
In \textit{M. Gordina} [Potential Anal. 12, No. 4, 325--357 (2000, Zbl 0960.46026)], the author had constructed the heat kernel measure \(\mu_t\) on the Hilbert-Schmidt orthogonal group \(\text{SO}_{\text{HS}}\), embedded in the space of Hilbert-Schmidt operators on an infinite-dimensional Hilbert space; then she had studied Hilbert spaces of holomorphic functions over \(\text{SO}_{\text{HS}}\). At the end of the article, she had claimed that her methods can be applied in more general situations. The aim of the present article is to verify this claim: The author generalizes her previous work to a setting which also includes examples like the Hilbert-Schmidt complex symplectic group \(\text{Sp}_{\text{HS}}\) or a group \(G\) of diagonal (infinite) complex matrices \(\text{diag}(1+ a_1,\dots, 1+ a_i,\dots)\), where \(\sum^\infty_{i=1}| a_i|^2< \infty\), \(a_i\neq 1\). From the author's abstract: ``The heat kernel measure \(\mu_t\) is constructed on \(\text{GL}(H)\), the group of invertible operators on a complex Hilbert space \(H\). This measure is determined by an infinite-dimensional Lie algebra \({\mathfrak g}\) and a Hermitian inner product on it. The Cameron-Martin subgroup \(G_{\text{CM}}\) is defined and its properties are discussed. In particular, there is an isometry from the \(L^2_{\mu_t}\)-closure of holomorphic polynomials into a space \({\mathcal H}^t(G_{\text{CM}})\) of functions holomorphic on \(G_{\text{CM}}\). This means that any element from this \(L^2_{\mu_t}\)-closure of holomorphic polynomials has a version holomorphic on \(G_{\text{CM}}\). In addition, there is an isometry from \({\mathcal H}^t(G_{\text{CM}})\) into a Hilbert space associated with the tensor algebra over \({\mathfrak g}\). The latter isometry is an infinite-dimensional analog of the Taylor expansion.''
scientific article; zbMATH DE number 1423459