# Closed graph theorem for cones?

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 ?

• 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
– erz
May 6, 2020 at 19:12
• @erz yes, of course you are right May 6, 2020 at 19:34