A non-uniform Datko–Pazy theorem for bounded operator semigroups

We present a non-uniform analogue of the classical Datko–Pazy theorem. Our main result shows that an integrability condition imposed on orbits originating in a fractional domain of the generator (as opposed to all orbits) implies polynomial stability of a bounded $C_0$-semigroup. As an application of this result we establish polynomial stability of a semigroup under a certain non-uniform Lyapunov-type condition. We moreover give a new proof, under slightly weaker assumptions, of a recent result deducing polynomial stability from a certain non-uniform observability condition.