Exterior product bundle over complex abstract Wiener space
A complex abstract Wiener space is the triplet \((B,H,\mu)\) of a complex separable Banach space \(B\), a complex separable Hilbert space \(H\) which is densely and continuously imbedded in \(B\) and a Borel probability measure \(\mu\) on \(B\) such that \[ \int_ B \exp(i\cdot \text{Re} \langle z,\varphi\rangle) \mu(dz)= \exp (-1/4 \|\varphi\|_{H^*}^ 2), \qquad \varphi\in B^*\subset H^*, \quad z\in B, \] and \(B^*\subset \bigcap_{n=1}^ \infty \text{Dom}(A^ n)\) for a strictly positive selfadjoint operator \(A\) on \(H^*\). \(B\) is regarded as an infinite dimensional manifold with cotangent space \((H_ R^*)^ c\) on each \(z\in B\). Consequently its exterior product bundle becomes \(B\times \Lambda^{p,q} (H_ R^*)^ c\) and the space of its \(L^ 2\)-sections becomes \(\Lambda^{p,q} (B)= L^ 2(B,\mu)\otimes \Lambda^{p,q} (H_ R^*)^ c\). Let \({\mathfrak H}^{p,q} (B)\) be the \(\overline{\partial}\)- cohomology group, \({\mathfrak h}_ A^{p,q}\) be the space of harmonic \((p,q)\)-forms. The author shows that the de Rham-Hodge-Kodaira's decomposition for \(\Lambda^{p,q}(B)\) holds and from this he concludes that \({\mathfrak H}^{p,q}(B)= {\mathfrak h}_ A^{p,q}\).
\(\overline{\partial}\)- cohomology group
scientific article; zbMATH DE number 219871