Causal invertibility

It is well known that in a given resolution space structure the inverse of a memoryless operator, when it exists, is also memoryless. Given a Hilbert resolution space, we define new resolution structures which preserve causality. We show that every causal, causally invertible operator can be viewed as a memoryless operator between resolution spaces. The characterization is both necessary and sufficient. We also identify causal operators which can be considered as memoryless in a suitable resolution space.