You should use a car seat on the plane. You don't drive your car with your child not in a carseat - why would you do it on a plane? If you hit any type of turbulence your child will not be protected in a plane seat. The adult seat belt could hurt internal organs. This is the same reason you do not use an adult seat belt on a child in a car.
To the original poster:
I have a sit'n stroll on order for my child. I'll be taking that along with an umbrella stroller on our upcoming cruise. I'd rather stroll my child thru the airport than carry a car seat! I'll check the umbrella stroller.