  The concept of hypocoercivity for linear evolution equations with dissipation
is discussed and equivalent characterizations that were developed for the
finite-dimensional case are extended to separable Hilbert spaces. Using the
concept of a hypocoercivity index, quantitative estimates on the short-time and
long-time decay behavior of a hypocoercive system are derived. As a useful tool
for analyzing the structural properties, an infinite-dimensional staircase form
is also derived and connections to linear systems and control theory are
presented. Several examples illustrate the new concepts and the results are
applied to the Lorentz kinetic equation.
