A detailed proof of the following criterion of causality in the frequency domain is presented: a linear time invariant (possibly unstable) system P mapping a subspace of the Hardy space H2into H2is causal if and only if for any x( · )ϵ Dom (P), exp (α) x( ·) ϵ H2for α > 0 implies exp(α) Px( · ) ϵ H2. Some applications of this criterion and properties of the causal systems are also presented