On the spectrum determined growth assumption and the perturbation ofC ₀ semigroups

Abstract

The spectrum determined growth property ofC ₀ semigroups in a Banach space is studied. It is shown that ifA generates aC ₀ semigroup in a Banach spaceX, which satisfies the following conditions: 1) for any σ>s(A), sup{‖R(λ;A)‖ | Reλ≥σ}<∞; 2) there is a ρ₀>ω(A) such that$\int_{ - \infty }^{ + \infty } {\left\| {R(\sigma _0 + i\tau ;A)x} \right\|^2 } d\tau< \infty$, ∀x∈X, and$\int_{ - \infty }^{ + \infty } {\left\| {R(\sigma _0 + i\tau ;A^* )f} \right\|^2 } d\tau< \infty$, ∀f∈X ^*, then ω(A=s(A). Moreover, it is also shown that ifA=A ₀+B is the infinitesimal generator of aC ₀ semigroup in Hilbert space, whereA ₀ is a discrete operator andB is bounded, then ω(A)=s(A). Finally the results obtained are applied to wave equation and thermoelastic system.