I always said, cheerfully, "hope you have a good day" when I left for work, expecting nothing back.
When I came home, "Hi, anything to check in on with the house, kids, pets, etc?" If she had something, i did full attention listening, validating, etc.
If not, then "Okay, if anything comes up, let me know, I'll be in the garage, doing "X', etc"
Then I would go about my business.
Whenever SHE would approach me on something, I stopped what I was doing, gave full attention (put the book down, turned away from the computer, etc) and gave full attention listening. If I was in the middle of something that I couldn't turn away from, say, like cooking, I said, "I need to get this saute done, and then I would be able to give you my full attention, does that work for you?"
Once or twice a week I would invite her to participate in something, like "I'm taking the dog for a walk, wanna come with?" If no, then "Okay, no problem, be back in a bit"
Sometimes, if it was realistic, I would approach her for assistance with something, BUT....it had to be something that I couldn't do on my own without help. She saw right through that (yes, I tried it....)
I think you get the idea...thing is to be sure you have no expectations, be pleasant, or, at least cordial. And her saying no is "no big deal to you"...like if she was a roommate. AND, GET BUSY in your OWN business, don't ask repeatedly, check back later, etc... one and done.
RH dubbed this "Aloof, but available", it is my mantra.
You have to keep aware of her mood, etc, and adjust as necessary, it will fluctuate. There'd be times when I wouldn't ask to do something, or skip the above because the chorus of demons scurrying about her head and the room told me..."uh, NOT a good time there T2, try back later"
In the depths of winter, I finally learned that within me there lay an invincible summer. - Albert Camus
Uncertainty is the very condition which impels people to unfold their powers.-Eric Fromm