A stochastic Datko-Pazy theorem

Let $H$ be a Hilbert space and $E$ a Banach space. In this note we present a sufficient condition for an operator $R:H\to E$ to be $\gamma$–radonifying in terms of Riesz sequences in $H$. This result is applied to recover a result of Lutz Weis and the second named author on the $R$-boundedness of resolvents, which is used to obtain a Datko-Pazy type theorem for the stochastic Cauchy problem. We also present some perturbation results.