I just ran an experiment and the results somewhat startled me. Both HA and US fly daily nonstops between HNL and PHX. HA flies a 767 and US flies a 757. In the past I have found HA to be cheaper.
Today I just picked two dates from thin air and found, using Orbitz, that ** UA ** has the cheapest nonstop fare for the two dates (4/27 & 4/30) at $365.00 (including taxes) using US flights 20 and 23 as code share flights. US came in at $420.00 for the same flights and the HA roundtrip came in at $498.00. HA's fare does include Lunch or Dinner, depending on direction and it is not an overnight flight from HNL, as is the US flight.
Frankly I expected HA to be cheaper and it may, or may not, be and I just picked a date when the load factor on HA was higher. I will say that I have flown that HA flight and it was far more comfortable in coach than a 757. However I doubt that US wants to add more capacity to the route and potentially lower yield if those seats can't sell at higher fares.