I am not sure what you mean by the "luggage compartment". Do you mean your cat will have to be in cargo?
At any rate, I recommend www.jetpets.com. I brought my cat from Australia to the US through JetPets. They are great people, will answer all your questions, and take great care of your cat.
It costs quite a bit (it was about $1200 for my cat), but is worth the price. They will come and get your cat, provide the container for the cat and take care of it every step of the trip.
I was worried that my cat would get stuck in a freezing cargo, without food or water, or would be next to tiger or something awful like that. They don't let that happen. Your cat will have water and JetPets checks out that no other animals that would stress out your cat will be in cargo.
Every airport the cat arrives to, Jetpets has a person waiting for your pet's arrival.
Also, when my cat arrived in the US, she was inspected by a veterinarian, was fed, bathed, and had time to run around and exercise.
When she got here, she smelled great!
JetPets are very communicative as well. If you are anxious and want to know where your cat is, just give them a call. They will tell you how things are going.
I highly recommend JetPets.
Good luck!
PS: I respectfully disagree with one of the other posters who said they would not allow their pet to travel in cargo. I tried desperately to find an airline that would allow me to have my cat travel with me in a carrier under the seat in front of mine, but no international airlines would do it. It is allowed within the United States on some airlines, but overseas flights simply will not do it.
Also, do not give the pet any tranquilizers. I cannot remember the exact reason, but at high altitudes, it can hurt your pet. JetPets can give you information about this.
By the way, in your pet's carrier, put a t-shirt that you have worn about a week or so in with your cat. It gives them the smell of you and makes them calmer.
Again, good luck!