A multistage compressor for a turbomachine, in particular an airplane turboprop or turbojet, the compressor comprising a double-walled casing having an inner wall made up of shrouds surrounding respective annular rows of moving blades and annular rows of straightening stator vanes, said shrouds being connected to the outer wall of the casing by independent suspension means enabling the radial clearances between the outer ends of the moving blades and the shrouds of the inner wall of the casing to be adjusted independently from one compression stage to another.