Decomposition of bilinear forms as sums of bounded forms
type: Item
instance of: scholarly article
title: Decomposition of bilinear forms as sums of bounded forms
zbMATH Open document ID: 1389.47181
author: Mohamed Elmursi
published in: Journal of Operator Theory
publication date: 2016-12-01T00:00:00Z
Mathematics Subject Classification ID: 47L07
Mathematics Subject Classification ID: 46L07
zbMATH DE Number: 6658782
zbMATH Keywords: Hilbert space
zbMATH Keywords: bounded bilinear forms
MaRDI profile type: Publication
full work available at URL: jot.2015mar11.2074
OpenAlex ID: W2964141685
DOI: 10.7900/JOT.2015MAR11.2074
review text: Let \(H,K\) be a Hilbert spaces and let \(A,C\in B(H)\) and \(B,D \in B(K)\). A~result noted by \textit{Q.-H. Xu} [Duke Math. J. 131, No. 3, 525--574 (2006; Zbl 1129.46048)] states that, if \(u\) is a bilinear form on \(H \times K\) such that \(| u(x,y)| \leq \| A(x)\| \| B(y)\| +\| C(x)\| +\| D(y)\| \) for all \(x \in H\),\(y\in K\), then \(u\) can be decomposed as sum \(u = u_1+u_2\) of bilinear forms such that \(| u_1(x,y)| \leq \| A(x)\| \| B(y)\| \) and \(| u_2(x,y)| \leq \| C(x)\| \| D(y)\| \). In this interesting paper, the author gives complete proof of this result and goes on to give a necessary and sufficient condition in the case of finite dimensional Hilbert spaces, for the decomposition theorem to hold when there are \(n\) terms in the bound for \(u\). This is then used to given an example of a \(u\) where there are three terms in the bound but \(u\) is not decomposable as a sum of \(3\)-forms, answering a question raised by Haagerup.
reviewed by: T.S.S.R.K. Rao
description: scientific article; zbMATH DE number 6658782