The AS brand is extremely strong in Southern California. Keep in mind that for years they were the only carrier flying LAX-DCA nonstop, as well as several LAX-Mexican resort destinations nonstop, and for a while became the second carrier to fly LAX-YYZ after AC (since been dropped).
Where the AS brand struggles is on the East Coast. I think their brand is strongest in the Baltimore-Washington area due to their historic presence in the region (DCA was their first East coast destination IIRC), and even today they have more flights from SEA to WAS than any other East Coast destination. They retrenched a bit in the market, having dropped BWI-LAX/SFO/SAN/PDX but are still strong IMO. Outside of WAS, NYC, and BOS, I would say their recognition isn't that great.
RJ85, F70, E135, E140, E145, E70, E75, E90, CR2, CR7, CR9, 717, 732, 733, 734, 735, 73G, 738, 739, 744ER, 752, 753, 762, 772, 77E, 77W, 789, 319, 320, 321, 332, 333, 343, 359, 388