  Realizing free semicircular elements on the full Fock space, we prove an
equivalence between rationality of operators obtained from them and finiteness
of the rank of their commutators with right annihilation operators. This is an
analogue of the result for the reduced $C^*$-algebra of the free group by G.
Duchamp and C. Reutenauer which was extended by PA. Linnel to densely defined
unbounded operators affiliated with the free group factor. Although their
result was motivated from quantized calculus in noncommutative geometry, we
state our results in terms of free probability theory.
