Backlog and Product Owner tasks

Hello everybody :)

I need your help for a simple question : Should we include the work of the Product Owner in the backlog?

The product owner spends time writing user stories, sometimes making models, writing specifications. Should we include them in the backlog? And if so, how? User story? Simple task? Should we also estimate them? Sounds consistent to me, but how? Planning poker, for example, makes sense as a team, but not alone.

Thanks a lot for your feedback :)