I had to look up what the hawk version of a falconer is, since it isn't a hawker. (It should have been, but that word took a weird turn... You expected me to make a tern joke there, but I'm not going to!... Oh, wait...)
It's an "austringer." (Which also took a turn, because most of the results are video-game-weapon-related. I searched for 'is there a hawk equivalent of a falconer' and got to the right word that way).
Our local BART system (regional electric trains) employs austringers to clear stations of pigeons, and it works. I asked the austringer directly, and it takes a while for the pigeons to take a hint. You can see a 'hawks at work' sign in the distance in the upper right corner of the photo, which I think helps commuters not ask silly questions about why austringers are wandering through the stations. Also, there is a BART anime mascot addressing this topic. (No, really.)
This particular austringer is (a) very handsome and (b) also has a hawk on his right arm in a stunningly gorgeous tattoo in the style of a Japanese painting. (I don't post photos of other people's tattoos without their permission, but I made giddy sounds when I reviewed the photo later. Such. Amazing. Ink.)
