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

Abstract

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)