Quasi-invariant measures on the orthogonal group over the Hilbert space
type: Item
instance of: scholarly article
title: Quasi-invariant measures on the orthogonal group over the Hilbert space
zbMATH Open document ID: 0546.28010
author: Hiroaki Shimomura
published in: Publications of the Research Institute for Mathematical Sciences, Kyoto University
publication date: 1984-01-01T00:00:00Z
review text: In this paper, three subjects in terms of translational-quasi-invariance are discussed for a probability measure \(\mu\) defined on the Borel field \({\mathcal B}\) derived from the strong topology on O(H). O(H) is the orthogonal group over the real Hilbert space with \(\dim (H)=\infty.\) Let G be a subgroup of O(H). First it is shown that ''there does not exist any \(\sigma\) -finite G-quasi-invariant measures'' on \({\mathcal B}\) as far as G acts transitively on the unit sphere of H.'' Next we choose a complete orthonormal system \(e_ 1,...,e_ n,..\). of H, and take G as a following set \(O_ o(H)=\cup^{\infty}_{n=1}O(n),\) where O(n) is a set of consisting of \(T\in O(H)\) which leaves \(e_ p\) invariant for all \(p>n\). Then \(''O_ o(H)\)-quasi-invariant probability measures certainly exist and the proof is carried out with the Schmidt's orthogonalization method''. This is the second subject. The last subject is an ergodic decomposition of \(O_ o(H)\)-quasi-invariant probability measures.
Mathematics Subject Classification ID: 28C20
Mathematics Subject Classification ID: 28C10
Mathematics Subject Classification ID: 60B05
zbMATH DE Number: 3869663
zbMATH Keywords: orthogonal group over the Hilbert space
zbMATH Keywords: translational-quasi-invariance
zbMATH Keywords: Schmidt's orthogonalization method
zbMATH Keywords: ergodic decomposition
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2977/prims/1195181831
OpenAlex ID: W2063821041
cites work: Q5512462
cites work: Q5533878
cites work: Invariant Integration Over the Infinite Dimensional Orthogonal Group and Related Spaces
cites work: On the construction of invariant measure over the orthogonal group on the Hilbert space by the method of Cayley transformation
cites work: Ergodic decomposition of quasi-invariant measures
cites work: Invariant measure of the infinite dimensional rotation group
DOI: 10.2977/PRIMS/1195181831
description: scientific article; zbMATH DE number 3869663