Compressor arrangement for generating compressed air using a motor-operator compressor unit, downstream of which at least one cooler unit is connected for cooling the generated compressed air, which enters the cooler unit via at least one inlet, flows there through a cooling structure, from a plurality of cooling passages connected in parallel and leaves the cooler unit via at least one outlet, wherein a permanently open bypass pipe is provided between the region of the inlet and the region of the outlet, the inner cross section of which bypass pipe is matched to the delivery capacity of the compressor unit and to the pressure difference between inlet and outlet such that the bypass pipe, at normal ambient temperatures, causes a higher flow resistance than the cooling structure, whereas, as freezing of the cooling structure progresses, the compressed air flows increasingly via the bypass pipe.