Hi Derek,
many thanks for your suggestions. I checked all.
Maybe I got a problem with my network constellation.
I installed the webadaptor in our DMZ. The portal and the Hosting Server are in the intranet.
The ports 6080, 6443, 7080 and 7443 between the webadaptor server and the portal/hosting server are open.
So the webadaptor server is not in the same network.
My ArcMap is in the intranet. So it can "see" the hosting server.
Now I don´t know how to go on.