Convexity of the Growth Bound of C 0-Semigroups of Operators

If e tA , t ≥ 0 is a C 0-semigroup of bounded linear operators in a Banach space X with infinitesimal generator A,then the growth bound of e tA , t ≥ 0, defined as a function of A, is ω 0(A) := \( {\lim _{t \to \infty }}\frac{1}{t}\log (|{e^{tA}}|) \) (see [4], p. 619). If B is in B(X) (the Banach algebra of bounded linear operators in X),then A + B is the infinitesimal generator of a C 0-semigroup e t (A+B), t ≥ 0 in X (see [7], p. 76). It is thus possible to consider ω 0(A+B) as a function of B∈B(X)and investigate its properties. The question we investigate here concerns the following property of the growth bound: If α ∈ (0,1) and B, C ∈ B(X), when is it true that $$ {\omega _0}(A + \alpha B + (1 - \alpha )C)\underline < \alpha {\omega _0}(A + B) + (1 - \alpha ){\omega _0}(A + C). $$ (1.1)