A characterization of rationality in free semicircular operators

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