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

We study decay rates for bounded $C_0$-semigroups from the perspective of $L^p$-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 $L^p$-infinite-time admissibility is provided for multiplication semigroups on $L^q$-spaces with $1\le q\le p<\mathrm{\infty }$. For polynomially stable $C_0$-semigroups on Hilbert spaces, we also give a sufficient condition for $L^2$-infinite-time admissibility.