PBA 30090
From United's Web-site
Carry-on baggage
For travel worldwide on United, you may carry on one bag and one personal item such as a purse, briefcase, or laptop computer.
A carry-on bag must fit under your seat or in the overhead bin.
Carry-on bag dimensions should not be more than 9 x 14 x 22 in/22 x 35 x 55 cm (length + width + height) or 45 linear in/112 linear cm (the length, height and width added together).
From American's Web-site
Our carry-on policy of one personal item (such as a purse or laptop bag) plus one bag remains in place. Current policies for baggage allowance as well as fees for checked baggage can be found below.
Trust me they may hassle you if you walk on with two bags and one of them is not a personal item. For what it is worth, I'm in that business....