Connection of hypocoercivity and hypocontractivity via the Cayley transform

The concepts of hypocoercivity and hypocontractivity and their relationship are studied for semi-dissipative continuous-time and discrete-time evolution equations in a Hilbert space setting. New proofs for the characterization of the short-time decay of the solution from the initial value are presented, that in particular characterize the constants in the leading terms of the solution when expanded in time. Maximally coercive/contractive representations of hypocoercive and hypocontractive semi-dissipative systems are presented, as well as the effect of different representations on the error estimates for the numerical solution.