[MortenTA]

I would play this more aggresive.

flop bet is fine
bet ~8 on turn
bet ~ 11 on river - I may c/c river

so which hands can he have?

Of course he could have a made flush on the flop but not likely.

KQ, KJ, KT, K9s, [Ad]x are the most likely holdings IMO.
You split with KQ but beat all the other holdings.

44 and 77 are possible but less likely than above.