On the iterates of the Laguerre operator

We use the iterates of the Laguerre operator to introduce Pilipovi\'c spaces on positive orthants. It is shown that such spaces coincide with $G-$type spaces g_α^α(Rᵈ₊) and G_α^α(Rᵈ₊), when α > 1, and α ≥ 1, respectively. However, in contrast to G-type spaces, Pilipovi\'c spaces on positive orthants are nontrivial below the critical index α = 1. We also remark that there is a natural isomorphism between subspaces of Pilipovi\'c spaces on Rᵈ consisting of even functions, and Pilipovi\'c spaces on positive orthants.