In this study, the structure of fractional spaces generated by the two-dimensional neutron transport operator A defined by formula Au = ω1 ∂u / ∂x + ω2 ∂u / ∂y is investigated. The positivity of A in C ( ℝ2 ) and Lp ( ℝ2 ) ; 1 ≤ p < ∞, is established. It is established that, for any 0 < α < 1 and 1 ≤ p < ∞; the norms of spaces Eα,p ( Lp ( ℝ2 ) ; A ) and Eα ( C ( ℝ2 ) ; A ) ; W p α ( ℝ2 ) and Cα ( ℝ2 ) are equivalent, respectively. The positivity of the neutron transport operator in Hölder space Cα ( ℝ2 ) and Slobodeckij space W p α ( ℝ2 ) is proved. In applications, theorems on the stability of Cauchy problem for the neutron transport equation in Hölder and Slobodeckij spaces are provided. © 2018 by the Tusi Mathematical Research Group.