Proprietorially is the adverb: it tells you how someone does the owning-attitude. They sit, touch, speak, or smile as if the place or the person is theirs. He smiled proprietorially at the garden. She draped a coat proprietorially over the best seat. The meaning is the same as proprietorial, only now it sits on the verb. Though actually, like the adjective, it often sounds a little too sure, a little too “this is mine.” Used of a person, it can feel unkind. Used of a room or a garden, it can be almost fond — still “as if I owned it,” but warmer. It is not a common everyday word; you will meet it more in writing than in chat.