Convexity and weak closedness of the image of operator balls under linear-fractional transformations
The unit ball consists of operators \(S\in L({\mathcal H}^+,{\mathcal H}^-)\), where \({\mathcal H}= {\mathcal H}^+\oplus{\mathcal H}^-\) is a Hilbert (Krein) space. The fractional function \((T_{21}+ T_{22}S)(T_{11}+ T_{12}S)^{-1}\) involves the blocks of \(T:{\mathcal H}\to{\mathcal H}\). Using a decomposition of the bistrict plus operators, the authors prove that the image of the unit ball through a fractional transformation is a convex and closed set in the weak operator topology.
scientific article; zbMATH DE number 1517709