Exterior product bundle over complex abstract Wiener space
type: Item
instance of: scholarly article
title: Exterior product bundle over complex abstract Wiener space
zbMATH Open document ID: 0796.46051
author: Takashi Nishimura
published in: Osaka Journal of Mathematics
publication date: 1993-06-29T00:00:00Z
review text: 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 \({\mathfrak H}^{p,q}(B)= {\mathfrak h}_ A^{p,q}\).
Mathematics Subject Classification ID: 46M20
Mathematics Subject Classification ID: 14F40
Mathematics Subject Classification ID: 58A12
Mathematics Subject Classification ID: 46G12
zbMATH DE Number: 219871
zbMATH Keywords: complex abstract Wiener space
zbMATH Keywords: strictly positive selfadjoint operator
zbMATH Keywords: exterior product bundle
zbMATH Keywords: \(L^ 2\)-sections
zbMATH Keywords: \(\overline{\partial}\)- cohomology group
zbMATH Keywords: space of harmonic \((p,q)\)-forms
zbMATH Keywords: de Rham-Hodge- Kodaira's decomposition
reviewed by: V. M. Deundjak
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 219871