On the spectral and growth bound of semigroups associated with hyperbolic equations

Some new and quite general conditions are presented to ensure equality of the spectral bound and the growth bound for those strongly continuous semigroups governing second-order (in time) equations in a Banach (or Hilbert) space. A counterexample arising from higher-order PDEs is given to show that the growth bound could be arbitrarily large while the spectral bound remains zero.