In a fluid flow machine, especially a gas turbine, axially penetrated by a hot gas stream, the rotor shaft is surrounded by heat shield segments or by base plates of the rotor blades on the rotor side. In the heat shield segments and in the base plates, cooling air chambers are provided, communicating with each other and with a cooling air source, which chambers communicate at least partially with gaps which extend in the longitudinal direction or peripheral direction of the rotor shaft between adjacent rotor blades or between heat shield segments and adjacent base plates of the rotor blades. These gaps are closed off by sealing bands such that a cooling air entry into the hot gas stream is only possible at the ends of the base plates of the rotor blades pointing contrary to the flow direction.