  For a contraction $C_0$-semigroup on a separable Hilbert space, the decay
rate is estimated by using the weak Poincar\'e inequalities for the symmetric
and anti-symmetric part of the generator. As applications, non-exponential
convergence rate is characterized for a class of degenerate diffusion
processes, so that the study of hypocoercivity is extended. Concrete examples
are presented.
