I'm having trouble understanding what you mean by not spending money, though. Obviously Liverpool, Arsenal, and United haven't needed new stadiums or anything since their American owners took over, so that's not really an issue. I can't speak for any rises in ticket prices or the like, but all of those mentioned have absolutely spent tons of money on players. I know there are a lot of smaller things to spend money on here and there as well, but I would think when most people think of owners spending money, it's #1 on players and #2 on facilities. The Americans have done that.
Heck, even Foley at Bournemouth has been investing in players since he came in last year.
If I have the info right, these teams have ownership that is all or partly American:
Arsenal
Aston Villa
Bournemouth
Burnley
Chelsea
Crystal Palace
Fulham
Liverpool
Manchester United
I'd say maybe Palace, Fulham, Burnley (though the owners are still relatively new) probably haven't invested a lot, but that's hardly a majority.