  Motivated by potential applications to partial differential equations, we
develop a theory of fine scales of decay rates for operator semigroups. The
theory contains, unifies, and extends several notable results in the literature
on decay of operator semigroups and yields a number of new ones. Its core is a
new operator-theoretical method of deriving rates of decay combining
ingredients from functional calculus, and complex, real and harmonic analysis.
It also leads to several results of independent interest.
