A Katznelson–Tzafriri Type Theorem for Almost Periodic Linear Evolution Equations

In this paper, we present an extension of a theorem due to Katznelson and Tzafriri to non-autonomous linear differential equations with almost periodic coefficients of the form x ′( t ) = A ( t ) x ( t ). To this end, we consider the evolution semigroup associated with the equation in a small invariant function space consisting of almost periodic functions with frequencies contained in the semi-module generated by the spectrum of A ( t ). The obtained (Perron type) conditions are stated in terms of the solvability of inhomogeneous equations associated with the equation under investigation.