There is a canonical form for a contractive operator from one Hilbert space to another; it is given by the polar resolution, together with the spectral theorem. If we seek a canonical form for a contractive operator together with a distinguished subspace of the domain space, we are bound to be disappointed. I will discuss the general form for this problem, explain why it doesn't quite qualify as "canonical" in the same sense, and show some wonderful things it can nevertheless do for us.