On idempotent operators on Fréchet spaces (Q792575)

The idempotent operators modulo-compact, that is the elements [D] of \({\mathcal L}(E,E)/{\mathcal K}(E,E)\) such that \([D]^ 2=[D],\) where \({\mathcal L}(E,E)\), (resp. \({\mathcal K}(E,E))\), denotes the space formed by the continuous, resp. compact, linear operators on E into E, E being a Fréchet space, are characterized. As a consequence of it, the complemented subspaces of \(E\times F\), cartesian product of two Fréchet spaces E and F such that (E,F)\(\in {\mathcal R}\) (all mappings on E into F are compact) are determined.