On Consistent Operators and Reflexivity

We study Hilbert space operators $${A=\oplus_{i \in \mathbb{N} } A_i}$$ which are consistent in the sense that each A i+1 contains a copy of A i . The formal definition is reminiscent of the classical ordering on projections in a von Neumann algebra. It is shown that if the powers of A are simultaneously consistent, then A must be reflexive. This is applied to study reflexivity of power partial isometries.