At least for Hyatt, the rooms are called "Respire" and as per their website:
"Respire by Hyatt rooms undergo an additional six-step process to reduce airborne particles and minimize the presence of potential irritants. This process includes installation of a state-of-the-art air purification system and treatment of all fabrics and surfaces in the guestroom. All Respire by Hyatt rooms are installed and maintained on an ongoing basis by PURE Solutions NA.These rooms (regardless of the hotel) are implemented by New York–based company Pure Solutions.
The Respire by Hyatt – Hypo-Allergenic Rooms purification process includes:
• Application of PURE Shield and PURE Clean to minimize presence of allergens on carpet, upholstery and other surfaces in the room.
• One-time shock treatment to minimize irritants.
• Complete disinfection of Air Handling Unit and Installation of removable tea tree oil cartridge to maintain these conditions.
• Installation of hypo-allergenic mattress & pillow encasings.
• Installation of a powerful air purifier by Healthway, listed as a Class II Medical device by the F.D.A.
• Recertification every six months."
Read more about what exactly is done to these rooms here.