Question about my new puppy and one year old dog...?

My puppy is 8 weeks old, and our older, one year old dog came with us to pick her out. The two got along immediately, but I've found that now, about a week in, the older dog will occasionally "pick fights" with the puppy, going over and stealing her chew toy or nipping her on the tail or deliberately jumping on her- the puppy immediately gets defensive and growls and nips. We supervise the two when they are together and there's a definite difference between play and fighting. The two dogs have separate areas: our puppy has taken over the small office that's attached to our kitchen and the older dog has the family room (also attached to the kitchen but on the opposite side of the room). The dogs have their own crates in their own areas as well as their own food that we keep in their respective areas. The two dogs cannot access each others space without supervision and the only time they interact is in the kitchen. What can I do to show my older dog that this younger puppy is not a threat?