\((DN)\)-\((\Omega)\)-type conditions for Fréchet operator spaces (Q1951190)

\((DN)\)- and \((\Omega)\)-type conditions have proved to be of central importance for the modern structure theory of Fréchet spaces (e.g., for the Vogt/Wagner characterization of subspaces and quotients of the space \((s)\) of rapidly decreasing sequences) and for several analytical problems like splitting of exact sequences or solvability of vector valued linear equations. In the present, interesting paper, the author accordingly develops the notions \((oDN)\) and \((o\Omega)\) adapted to the setting of Fréchet operator spaces. The inheritance of these notions from the underlying space to several quantizations is studied in detail: for any Fréchet space \(X\), the minimal and the maximal quantizations satisfy \((oDN)\) and \((o\Omega)\) if \(X\) satisfies \((DN)\) (and \((\Omega)\), respectively). For Fréchet-Hilbert spaces, this is also true for the row, column and Pisier quantizations. Notice that the quantization is unique up to a complete isomorphism if \(X\) is a nuclear Fréchet space. Finally, a description of \((oDN)\) and \((o\Omega)\) using polars is provided.