The longstanding problems of the linear stability of plane Couette flow and circular pipe flow (to axisymmetric disturbances) are solved by operator theory. It is shown simply that both are stable for all Reynolds numbers and wave numbers. The proof is based on the von Neumann extension of a semi-bounded symmetric operator and the notion of a square root of an unbounded positive definite selfadjoint operator. The use of the latter operator representation is new for this type of hydrodynamic stability problem. It is made clear how the method will apply in other problems with a similar structure such as the planar stability of Couette flow between rotating coaxial cylinders and parabolic Poiseuille flow.