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

Abstract

Let $\mathcal {U}=\{U(t,s)\}_{t\ge s\ge 0}$ be a strongly continuous and exponentially bounded evolution family acting on a complex Banach space X and let $\mathcal {X}$ be a certain Banach function space of X-valued functions. We prove that the growth bound of the family $\mathcal {U}$ is less than or equal to $-\frac{1}{c(\mathcal {U}, \mathcal {X})}$ provided that the convolution operator $f\mapsto \mathcal {U}*f$ acts on $\mathcal {X}.$ It is well known that under the latter assumption, the convolution operator is bounded and then $c(\mathcal {U}, \mathcal {X})$ denotes (ad-hoc) its norm in $\mathcal {L}(\mathcal {X}).$ As a consequence, we prove that if $\sup \nolimits _{s\ge 0}\int \nolimits _{s}^\infty \Vert U(t,s)\Vert dt=u_1(\mathcal {U})<\infty ,$ then $\omega _0(\mathcal {U})u_1(\mathcal {U})\le -1.$ Finally, we give an example showing that the accuracy of the estimates may be quite accurate.