Two Banach space methods and dual operator algebras
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.
convexity and dominancy in Banach spaces
scientific article; zbMATH DE number 4080134
scientific article