On the spectrum determined growth assumption and the perturbation of C-0 semigroups

The spectrum determined growth property of C-0 semigroups in a Banach space is studied. It is shown that if A generates a C-0 semigroup in a Banach sp ace X, which satisfies the following conditions: 1) for any sigma > s(A), s up{\\R(lambda; A)\\ / Re lambda greater than or equal to sigma} < <infinity >; 2) there is a sigma (0) > omega (A) such that integral (+infinity)(-infi nity+) \\R(sigma (0) + i tau; A)x\\(2)d tau < <infinity>, For Allx is an el ement of X, and integral (+infinity)(-infinity) \\R(sigma (0) + i tau; A*)f \\2d tau < <infinity>, For Allf is an element of X*, then omega (A) = s(A). Moreover, it is also shown that if A = A(0) + B is the infinitesimal gener ator of a C-0 semigroup in Hilbert space, where A(0) is a discrete operator and B is bounded, then omega (A) = s(A). Finally the results obtained are applied to wave equation and thermoelastic system.