The fact that she chases him off makes it seem more like play aggression than actual aggression, or at least that she’s not being harmed by it. If he had her running scared all the time, it would be more concerning. Cats make a lot of noise when they fight, and it can be hard to know if it’s real or all in fun. Has he ever actually hurt her or drawn blood?
DH and I each had two cats when he moved in with my family early in our relationship. “My” (family pets) two boys tolerated one of his two girls, but one of my boys and one of his girls did NOT get along, to the point we had to keep them separated when we were not able to supervise them. My boys got along well most of them time, although they did fight on occasion that seemed to be mostly either in fun or occasional sibling rivalry. Three of the cats have since passed. Even when we were down to just two, they didn’t get along.
DH never believed me that it was HIS cat that would start every.one. of the fights. Seriously, my cat would be on the couch between us while we watched TV, his cat would jump up on his lap and be fine until she realized my cat was there, (and he would be totally ignoring her), and then would flip out growling and hissing. My boy would just look at her like, “What is your prooblem?” Hehehe…she wasn’t so bright, but DH loved her, (my cat is a Maine Coon, so he was a lot bigger than her, and he was more than capable of winning any fight she wanted to pick, but she still tried).
After DH’s cat died, we fostered a female cat, (and later adopted her), and they got along fne. Not to the point of snuggling, playing, or mutual grooming, but slightly beyond tolerating each other, (they would sleep on the guest bed near each other, etc). Finally proved my point that it was DH’s cat causing the problem, (he always insisted my cat was just a jerk!). If we hadn’t had to leave my boy behind when we moved out, (he stayed with my parents since he was a family pet, and a senior with developing hip problems that would make our new home difficult for him), they may have eventually gotten to the next level of kitty friendship.
Anyway, I’ve realized I just rambled quite a bit…but the point I was trying to make is that cats don’t have to be best friends to coexist. As another poster said, the species would normally live alone, only coming together for reproduction. When humans make cats live together, they will usually develop a social heirarchy, (hence the “dirty look” your male sometimes gives your female when she’s getting too much of “his” attention). Just like humans, they will have arguments now and then. Sometimes they like to fight for the fun of it, (think of siblings wrestling). If your cats can be best friends, that’s awesome, but it doesn’t always happen, and that’s okay, too. Your female doesn’t seem like she’s being terrorized, (bossed around a bit, yes, but it doesn’t sound like she’s living in fear of your male and his attacks). If you are truly concerned, watch some “My Cat From Hell” episodes for some great ideas on helping promote harmony in your home. But just remember, NOBODY gets along 100% of the time.