In the paper "A strong open mapping theorem for surjections from cones onto Banach spaces, Marcel de Jeu and Miek Masserschmidt, Adv. Math." it is proved (among other things) that if $X, Y$ are complex Banach spaces, $C\subset X$ is a closed cone and $T:C\mapsto X$ is a continuous additive posotively homogeneous and surjective map, then it is open.

As far as I can see, if one follows the proof of the closed graph theorem from the open mapping mutatis mutandis, one can arrive at a " closed graph theorem for cones". I.e. for $X,Y,C$ as before and $T:C\subset Y \mapsto X$ is additive, positively homegeneous and has closed graph (in the topology of $X \times Y$) then there exists $K>0$ such that $\Vert T c\Vert \leq K \Vert c \Vert, \,\,\, \forall c \in C $. Although I am bit suspicious because nothing like this is mentioned in the paper.

Am I missing something ?

  • 1
    $\begingroup$ But then is you have a projection from the graph in $X\times Y$ onto $C$, not the whole space, so you cannot conclude that it is invertible $\endgroup$
    – erz
    May 6, 2020 at 19:12
  • $\begingroup$ @erz yes, of course you are right $\endgroup$ May 6, 2020 at 19:34


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