Decay of Operator Semigroups, Infinite-time Admissibility, and Related Resolvent Estimates

Abstract: We study decay rates for bounded $C_0$-semigroups from the perspective of $L<sup>p$-infinite-time admissibility and related resolvent estimates. In the Hilbert space setting, polynomial decay of semigroup orbits is characterized by the resolvent behavior in the open right half-plane. A similar characterization based on $L<sup>p$-infinite-time admissibility is provided for multiplication semigroups on $L<sup>q$-spaces with $1 \leq q \leq p &lt; \infty$. For polynomially stable $C_0$-semigroups on Hilbert spaces, we also give a sufficient condition for $L<sup>2$-infinite-time admissibility.