An inequality concerning the growth bound of an evolution family and the norm of a convolution operator

Let U= { U(t, s) } _{t ≥ s ≥ 0} be a strongly continuous and exponentially bounded evolution family acting on a complex Banach space X and let X be a certain Banach function space of X-valued functions. We prove that the growth bound of the family U is less than or equal to -1c(U,X) provided that the convolution operator f↦ U∗ f acts on X. It is well known that under the latter assumption, the convolution operator is bounded and then c(U, X) denotes (ad-hoc) its norm in L(X). As a consequence, we prove that if sups≥0∫s∞‖U(t,s)‖dt=u1(U)<∞, then ω₀(U) u₁(U) ≤ - 1. Finally, we give an example showing that the accuracy of the estimates may be quite accurate.