Two Banach space methods and dual operator algebras
type: Item
instance of: scholarly article
title: Two Banach space methods and dual operator algebras
zbMATH Open document ID: 0661.46004
DOI: 10.1016/0022-1236(88)90122-X
author: H. Bercovici
author: Ciprian Foias
author: C. Pearcy
published in: Journal of Functional Analysis
publication date: 1988-01-01T00:00:00Z
review text: Let \({\mathcal X}\), \({\mathcal Y}\), and \({\mathcal Z}\) be Banach spaces, let \({\mathcal D}\subset {\mathcal X}\times {\mathcal Y}\), and let T:\({\mathcal D}\to {\mathcal Z}\) be a bilinear map. It is known that the analog of the Banach's open mapping theorem for a surjective T is not true in general. In this article the authors investigate the richtness of the set \(B=\{T(x,y):(x,y)\in {\mathcal D}\), \(\| x\| \leq 1,\| y\| \leq 1,\| T(x,y)\| \leq M\}\), \(M>0\), under extra conditions on T. First a new type of convexity and dominancy in Banach spaces is introduced and sufficient conditions in order that \(\bar B\) contains a given convex set are derived. Next, using another approach when \({\mathcal X}\) and \({\mathcal Y}\) are Hilbert spaces, sufficient conditions for T to be open are proved. An interesting application of these is given for solvability of systems \(T(x_ i,y_ j)=z_{ij}\), \(0\leq i\), \(j<\infty\). Finally, an application for contraction of class \(C_{00}\) is given.
Mathematics Subject Classification ID: 46A30
Mathematics Subject Classification ID: 47L50
zbMATH DE Number: 4080134
zbMATH Keywords: bilinear map
zbMATH Keywords: Banach's open mapping theorem
zbMATH Keywords: convexity and dominancy in Banach spaces
zbMATH Keywords: solvability of systems
reviewed by: Ali-Akbar Jafarian
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4080134